Derivative

A point where the second derivative of a function changes sign is called an inflection point.[2] At an inflection point, the second derivative may be zero, as in the case of the inflection point x = 0 of the function given by ${\displaystyle f(x)=x^{3}}$, or it may fail to exist, as in the case of the inflection point x = 0 of the function given by ${\displaystyle f(x)=x^{\frac {1}{3}}}$. At an inflection point, a function switches from being a convex function to being a concave function or vice versa.

The symbols ${\displaystyle dx}$, ${\displaystyle dy}$, and ${\displaystyle {\frac {dy}{dx}}}$ were introduced by Gottfried Wilhelm Leibniz in 1675.[3] It is still commonly used when the equation y = f(x) is viewed as a functional relationship between dependent and independent variables. Then the first derivative is denoted by

${\displaystyle {\frac {dy}{dx}},\quad {\frac {df}{dx}},{\text{ or }}{\frac {d}{dx}}f,}$

and was once thought of as an infinitesimal quotient. Higher derivatives are expressed using the notation

${\displaystyle {\frac {d^{n}y}{dx^{n}}},\quad {\frac {d^{n}f}{dx^{n}}},{\text{ or }}{\frac {d^{n}}{dx^{n}}}f}$

for the nth derivative of ${\displaystyle y=f(x)}$. These are abbreviations for multiple applications of the derivative operator. For example,

${\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {d}{dx}}\left({\frac {dy}{dx}}\right).}$

With Leibniz's notation, we can write the derivative of ${\displaystyle y}$ at the point ${\displaystyle x=a}$ in two different ways:

${\displaystyle \left.{\frac {dy}{dx}}\right|_{x=a}={\frac {dy}{dx}}(a).}$

Leibniz's notation allows one to specify the variable for differentiation (in the denominator), which is relevant in partial differentiation. It also can be used to write the chain rule as[Note 2]

${\displaystyle {\frac {dy}{dx}}={\frac {dy}{du}}\cdot {\frac {du}{dx}}.}$

Sometimes referred to as prime notation,[4] one of the most common modern notations for differentiation is due to Joseph-Louis Lagrange and uses the prime mark, so that the derivative of a function ${\displaystyle f}$ is denoted ${\displaystyle f'}$. Similarly, the second and third derivatives are denoted

${\displaystyle (f')'=f''}$   and   ${\displaystyle (f'')'=f'''.}$

To denote the number of derivatives beyond this point, some authors use Roman numerals in superscript, whereas others place the number in parentheses:

${\displaystyle f^{\mathrm {iv} }}$   or   ${\displaystyle f^{(4)}.}$

The latter notation generalizes to yield the notation ${\displaystyle f^{(n)}}$ for the nth derivative of ${\displaystyle f}$ – this notation is most useful when we wish to talk about the derivative as being a function itself, as in this case the Leibniz notation can become cumbersome.

Newton's notation for differentiation, also called the dot notation, places a dot over the function name to represent a time derivative. If ${\displaystyle y=f(t)}$, then

${\displaystyle {\dot {y}}}$   and   ${\displaystyle {\ddot {y}}}$

denote, respectively, the first and second derivatives of ${\displaystyle y}$. This notation is used exclusively for derivatives with respect to time or arc length. It is typically used in differential equations in physics and differential geometry.[5][6] The dot notation, however, becomes unmanageable for high-order derivatives (order 4 or more) and cannot deal with multiple independent variables.

Euler's notation uses a differential operator ${\displaystyle D}$, which is applied to a function ${\displaystyle f}$ to give the first derivative ${\displaystyle Df}$. The nth derivative is denoted ${\displaystyle D^{n}f}$.

If y = f(x) is a dependent variable, then often the subscript x is attached to the D to clarify the independent variable x. Euler's notation is then written

${\displaystyle D_{x}y}$   or   ${\displaystyle D_{x}f(x)}$,

although this subscript is often omitted when the variable x is understood, for instance when this is the only independent variable present in the expression.

Euler's notation is useful for stating and solving linear differential equations.

Here are the rules for the derivatives of the most common basic functions, where a is a real number.

• Derivatives of powers:

  ${\displaystyle {\frac {d}{dx}}x^{a}=ax^{a-1}.}$

• Exponential and logarithmic functions:

  ${\displaystyle {\frac {d}{dx}}e^{x}=e^{x}.}$

  ${\displaystyle {\frac {d}{dx}}a^{x}=a^{x}\ln(a),\qquad a>0}$

  ${\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}},\qquad x>0.}$

  ${\displaystyle {\frac {d}{dx}}\log _{a}(x)={\frac {1}{x\ln(a)}},\qquad x,a>0}$

• Trigonometric functions:

  ${\displaystyle {\frac {d}{dx}}\sin(x)=\cos(x).}$

  ${\displaystyle {\frac {d}{dx}}\cos(x)=-\sin(x).}$

  ${\displaystyle {\frac {d}{dx}}\tan(x)=\sec ^{2}(x)={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x).}$

• Inverse trigonometric functions:

  ${\displaystyle {\frac {d}{dx}}\arcsin(x)={\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1.}$

  ${\displaystyle {\frac {d}{dx}}\arccos(x)=-{\frac {1}{\sqrt {1-x^{2}}}},\qquad -1<x<1.}$

  ${\displaystyle {\frac {d}{dx}}\arctan(x)={\frac {1}{1+x^{2}}}}$

Here are some of the most basic rules for deducing the derivative of a compound function from derivatives of basic functions.

• Constant rule: if f(x) is constant, then

  ${\displaystyle f'(x)=0.}$

• Sum rule:

  ${\displaystyle (\alpha f+\beta g)'=\alpha f'+\beta g'}$ for all functions f and g and all real numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$.

• Product rule:

  ${\displaystyle (fg)'=f'g+fg'}$ for all functions f and g. As a special case, this rule includes the fact ${\displaystyle (\alpha f)'=\alpha f'}$ whenever ${\displaystyle \alpha }$ is a constant, because ${\displaystyle \alpha 'f=0\cdot f=0}$ by the constant rule.

• Quotient rule:

  ${\displaystyle \left({\frac {f}{g}}\right)'={\frac {f'g-fg'}{g^{2}}}}$ for all functions f and g at all inputs where g ≠ 0.

• Chain rule for composite functions: If ${\displaystyle f(x)=h(g(x))}$, then

  ${\displaystyle f'(x)=h'(g(x))\cdot g'(x).}$

The derivative of the function given by

${\displaystyle f(x)=x^{4}+\sin \left(x^{2}\right)-\ln(x)e^{x}+7}$

is

${\displaystyle {\begin{aligned}f'(x)&=4x^{(4-1)}+{\frac {d\left(x^{2}\right)}{dx}}\cos \left(x^{2}\right)-{\frac {d\left(\ln {x}\right)}{dx}}e^{x}-\ln(x){\frac {d\left(e^{x}\right)}{dx}}+0\\&=4x^{3}+2x\cos \left(x^{2}\right)-{\frac {1}{x}}e^{x}-\ln(x)e^{x}.\end{aligned}}}$

Here the second term was computed using the chain rule and third using the product rule. The known derivatives of the elementary functions x², x⁴, sin(x), ln(x) and exp(x) = e^x, as well as the constant 7, were also used.

A vector-valued function y of a real variable sends real numbers to vectors in some vector space Rⁿ. A vector-valued function can be split up into its coordinate functions y₁(t), y₂(t), ..., y_n(t), meaning that y(t) = (y₁(t), ..., y_n(t)). This includes, for example, parametric curves in R² or R³. The coordinate functions are real valued functions, so the above definition of derivative applies to them. The derivative of y(t) is defined to be the vector, called the tangent vector, whose coordinates are the derivatives of the coordinate functions. That is,

${\displaystyle \mathbf {y} '(t)=(y'_{1}(t),\ldots ,y'_{n}(t)).}$

Equivalently,

${\displaystyle \mathbf {y} '(t)=\lim _{h\to 0}{\frac {\mathbf {y} (t+h)-\mathbf {y} (t)}{h}},}$

if the limit exists. The subtraction in the numerator is the subtraction of vectors, not scalars. If the derivative of y exists for every value of t, then y′ is another vector-valued function.

If e₁, ..., e_n is the standard basis for Rⁿ, then y(t) can also be written as y₁(t)e₁ + ⋯ + y_n(t)e_n. If we assume that the derivative of a vector-valued function retains the linearity property, then the derivative of y(t) must be

${\displaystyle y'_{1}(t)\mathbf {e} _{1}+\cdots +y'_{n}(t)\mathbf {e} _{n}}$

because each of the basis vectors is a constant.

This generalization is useful, for example, if y(t) is the position vector of a particle at time t; then the derivative y′(t) is the velocity vector of the particle at time t.

Suppose that f is a function that depends on more than one variable—for instance,

${\displaystyle f(x,y)=x^{2}+xy+y^{2}.}$

f can be reinterpreted as a family of functions of one variable indexed by the other variables:

${\displaystyle f(x,y)=f_{x}(y)=x^{2}+xy+y^{2}.}$

In other words, every value of x chooses a function, denoted f_x, which is a function of one real number.[Note 3] That is,

${\displaystyle x\mapsto f_{x},}$

${\displaystyle f_{x}(y)=x^{2}+xy+y^{2}.}$

Once a value of x is chosen, say a, then f(x, y) determines a function f_a that sends y to a² + ay + y²:

${\displaystyle f_{a}(y)=a^{2}+ay+y^{2}.}$

In this expression, a is a constant, not a variable, so f_a is a function of only one real variable. Consequently, the definition of the derivative for a function of one variable applies:

${\displaystyle f_{a}'(y)=a+2y.}$

The above procedure can be performed for any choice of a. Assembling the derivatives together into a function gives a function that describes the variation of f in the y direction:

${\displaystyle {\frac {\partial f}{\partial y}}(x,y)=x+2y.}$

This is the partial derivative of f with respect to y. Here ∂ is a rounded d called the partial derivative symbol. To distinguish it from the letter d, ∂ is sometimes pronounced "der", "del", or "partial" instead of "dee".

In general, the partial derivative of a function f(x₁, …, x_n) in the direction x_i at the point (a₁, ..., a_n) is defined to be:

${\displaystyle {\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n})=\lim _{h\to 0}{\frac {f(a_{1},\ldots ,a_{i}+h,\ldots ,a_{n})-f(a_{1},\ldots ,a_{i},\ldots ,a_{n})}{h}}.}$

In the above difference quotient, all the variables except x_i are held fixed. That choice of fixed values determines a function of one variable

${\displaystyle f_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}(x_{i})=f(a_{1},\ldots ,a_{i-1},x_{i},a_{i+1},\ldots ,a_{n}),}$

and, by definition,

${\displaystyle {\frac {df_{a_{1},\ldots ,a_{i-1},a_{i+1},\ldots ,a_{n}}}{dx_{i}}}(a_{i})={\frac {\partial f}{\partial x_{i}}}(a_{1},\ldots ,a_{n}).}$

In other words, the different choices of a index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.

This is fundamental for the study of the functions of several real variables. Let f(x₁, ..., x_n) be such a real-valued function. If all partial derivatives ∂f / ∂x_j of f are defined at the point a = (a₁, ..., a_n), these partial derivatives define the vector

${\displaystyle \nabla f(a_{1},\ldots ,a_{n})=\left({\frac {\partial f}{\partial x_{1}}}(a_{1},\ldots ,a_{n}),\ldots ,{\frac {\partial f}{\partial x_{n}}}(a_{1},\ldots ,a_{n})\right),}$

which is called the gradient of f at a. If f is differentiable at every point in some domain, then the gradient is a vector-valued function ∇f that maps the point (a₁, ..., a_n) to the vector ∇f(a₁, ..., a_n). Consequently, the gradient determines a vector field.

If f is a real-valued function on Rⁿ, then the partial derivatives of f measure its variation in the direction of the coordinate axes. For example, if f is a function of x and y, then its partial derivatives measure the variation in f in the x direction and the y direction. They do not, however, directly measure the variation of f in any other direction, such as along the diagonal line y = x. These are measured using directional derivatives. Choose a vector

${\displaystyle \mathbf {v} =(v_{1},\ldots ,v_{n}).}$

The directional derivative of f in the direction of v at the point x is the limit

${\displaystyle D_{\mathbf {v} }{f}(\mathbf {x} )=\lim _{h\rightarrow 0}{\frac {f(\mathbf {x} +h\mathbf {v} )-f(\mathbf {x} )}{h}}.}$

In some cases it may be easier to compute or estimate the directional derivative after changing the length of the vector. Often this is done to turn the problem into the computation of a directional derivative in the direction of a unit vector. To see how this works, suppose that v = λu where u is a unit vector in the direction of v. Substitute h = k/λ into the difference quotient. The difference quotient becomes:

${\displaystyle {\frac {f(\mathbf {x} +(k/\lambda )(\lambda \mathbf {u} ))-f(\mathbf {x} )}{k/\lambda }}=\lambda \cdot {\frac {f(\mathbf {x} +k\mathbf {u} )-f(\mathbf {x} )}{k}}.}$

This is λ times the difference quotient for the directional derivative of f with respect to u. Furthermore, taking the limit as h tends to zero is the same as taking the limit as k tends to zero because h and k are multiples of each other. Therefore, D_v(f) = λD_u(f). Because of this rescaling property, directional derivatives are frequently considered only for unit vectors.

If all the partial derivatives of f exist and are continuous at x, then they determine the directional derivative of f in the direction v by the formula:

${\displaystyle D_{\mathbf {v} }{f}({\boldsymbol {x}})=\sum _{j=1}^{n}v_{j}{\frac {\partial f}{\partial x_{j}}}.}$

This is a consequence of the definition of the total derivative. It follows that the directional derivative is linear in v, meaning that D_{v + w}(f) = D_v(f) + D_w(f).

The same definition also works when f is a function with values in R^m. The above definition is applied to each component of the vectors. In this case, the directional derivative is a vector in R^m.

When f is a function from an open subset of Rⁿ to R^m, then the directional derivative of f in a chosen direction is the best linear approximation to f at that point and in that direction. But when n > 1, no single directional derivative can give a complete picture of the behavior of f. The total derivative gives a complete picture by considering all directions at once. That is, for any vector v starting at a, the linear approximation formula holds:

${\displaystyle f(\mathbf {a} +\mathbf {v} )\approx f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {v} .}$

Just like the single-variable derivative, f ′(a) is chosen so that the error in this approximation is as small as possible.

If n and m are both one, then the derivative f ′(a) is a number and the expression f ′(a)v is the product of two numbers. But in higher dimensions, it is impossible for f ′(a) to be a number. If it were a number, then f ′(a)v would be a vector in Rⁿ while the other terms would be vectors in R^m, and therefore the formula would not make sense. For the linear approximation formula to make sense, f ′(a) must be a function that sends vectors in Rⁿ to vectors in R^m, and f ′(a)v must denote this function evaluated at v.

To determine what kind of function it is, notice that the linear approximation formula can be rewritten as

${\displaystyle f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} )\approx f'(\mathbf {a} )\mathbf {v} .}$

Notice that if we choose another vector w, then this approximate equation determines another approximate equation by substituting w for v. It determines a third approximate equation by substituting both w for v and a + v for a. By subtracting these two new equations, we get

${\displaystyle f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\approx f'(\mathbf {a} +\mathbf {v} )\mathbf {w} -f'(\mathbf {a} )\mathbf {w} .}$

If we assume that v is small and that the derivative varies continuously in a, then f ′(a + v) is approximately equal to f ′(a), and therefore the right-hand side is approximately zero. The left-hand side can be rewritten in a different way using the linear approximation formula with v + w substituted for v. The linear approximation formula implies:

${\displaystyle {\begin{aligned}0&\approx f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} +\mathbf {w} )+f(\mathbf {a} )\\&=(f(\mathbf {a} +\mathbf {v} +\mathbf {w} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {v} )-f(\mathbf {a} ))-(f(\mathbf {a} +\mathbf {w} )-f(\mathbf {a} ))\\&\approx f'(\mathbf {a} )(\mathbf {v} +\mathbf {w} )-f'(\mathbf {a} )\mathbf {v} -f'(\mathbf {a} )\mathbf {w} .\end{aligned}}}$

This suggests that f ′(a) is a linear transformation from the vector space Rⁿ to the vector space R^m. In fact, it is possible to make this a precise derivation by measuring the error in the approximations. Assume that the error in these linear approximation formula is bounded by a constant times ||v||, where the constant is independent of v but depends continuously on a. Then, after adding an appropriate error term, all of the above approximate equalities can be rephrased as inequalities. In particular, f ′(a) is a linear transformation up to a small error term. In the limit as v and w tend to zero, it must therefore be a linear transformation. Since we define the total derivative by taking a limit as v goes to zero, f ′(a) must be a linear transformation.

In one variable, the fact that the derivative is the best linear approximation is expressed by the fact that it is the limit of difference quotients. However, the usual difference quotient does not make sense in higher dimensions because it is not usually possible to divide vectors. In particular, the numerator and denominator of the difference quotient are not even in the same vector space: The numerator lies in the codomain R^m while the denominator lies in the domain Rⁿ. Furthermore, the derivative is a linear transformation, a different type of object from both the numerator and denominator. To make precise the idea that f ′(a) is the best linear approximation, it is necessary to adapt a different formula for the one-variable derivative in which these problems disappear. If f : R → R, then the usual definition of the derivative may be manipulated to show that the derivative of f at a is the unique number f ′(a) such that

${\displaystyle \lim _{h\to 0}{\frac {f(a+h)-(f(a)+f'(a)h)}{h}}=0.}$

This is equivalent to

${\displaystyle \lim _{h\to 0}{\frac {|f(a+h)-(f(a)+f'(a)h)|}{|h|}}=0}$

because the limit of a function tends to zero if and only if the limit of the absolute value of the function tends to zero. This last formula can be adapted to the many-variable situation by replacing the absolute values with norms.

The definition of the total derivative of f at a, therefore, is that it is the unique linear transformation f ′(a) : Rⁿ → R^m such that

${\displaystyle \lim _{\mathbf {h} \to 0}{\frac {\lVert f(\mathbf {a} +\mathbf {h} )-(f(\mathbf {a} )+f'(\mathbf {a} )\mathbf {h} )\rVert }{\lVert \mathbf {h} \rVert }}=0.}$

Here h is a vector in Rⁿ, so the norm in the denominator is the standard length on Rⁿ. However, f′(a)h is a vector in R^m, and the norm in the numerator is the standard length on R^m. If v is a vector starting at a, then f ′(a)v is called the pushforward of v by f and is sometimes written f_∗v.

If the total derivative exists at a, then all the partial derivatives and directional derivatives of f exist at a, and for all v, f ′(a)v is the directional derivative of f in the direction v. If we write f using coordinate functions, so that f = (f₁, f₂, ..., f_m), then the total derivative can be expressed using the partial derivatives as a matrix. This matrix is called the Jacobian matrix of f at a:

${\displaystyle f'(\mathbf {a} )=\operatorname {Jac} _{\mathbf {a} }=\left({\frac {\partial f_{i}}{\partial x_{j}}}\right)_{ij}.}$

The existence of the total derivative f′(a) is strictly stronger than the existence of all the partial derivatives, but if the partial derivatives exist and are continuous, then the total derivative exists, is given by the Jacobian, and depends continuously on a.

The definition of the total derivative subsumes the definition of the derivative in one variable. That is, if f is a real-valued function of a real variable, then the total derivative exists if and only if the usual derivative exists. The Jacobian matrix reduces to a 1×1 matrix whose only entry is the derivative f′(x). This 1×1 matrix satisfies the property that f(a + h) − (f(a) + f ′(a)h) is approximately zero, in other words that

${\displaystyle f(a+h)\approx f(a)+f'(a)h.}$

Up to changing variables, this is the statement that the function ${\displaystyle x\mapsto f(a)+f'(a)(x-a)}$ is the best linear approximation to f at a.

The total derivative of a function does not give another function in the same way as the one-variable case. This is because the total derivative of a multivariable function has to record much more information than the derivative of a single-variable function. Instead, the total derivative gives a function from the tangent bundle of the source to the tangent bundle of the target.

The natural analog of second, third, and higher-order total derivatives is not a linear transformation, is not a function on the tangent bundle, and is not built by repeatedly taking the total derivative. The analog of a higher-order derivative, called a jet, cannot be a linear transformation because higher-order derivatives reflect subtle geometric information, such as concavity, which cannot be described in terms of linear data such as vectors. It cannot be a function on the tangent bundle because the tangent bundle only has room for the base space and the directional derivatives. Because jets capture higher-order information, they take as arguments additional coordinates representing higher-order changes in direction. The space determined by these additional coordinates is called the jet bundle. The relation between the total derivative and the partial derivatives of a function is paralleled in the relation between the kth order jet of a function and its partial derivatives of order less than or equal to k.

By repeatedly taking the total derivative, one obtains higher versions of the Fréchet derivative, specialized to R^p. The kth order total derivative may be interpreted as a map

${\displaystyle D^{k}f:\mathbb {R} ^{n}\to L^{k}(\mathbb {R} ^{n}\times \cdots \times \mathbb {R} ^{n},\mathbb {R} ^{m})}$

which takes a point x in Rⁿ and assigns to it an element of the space of k-linear maps from Rⁿ to R^m – the "best" (in a certain precise sense) k-linear approximation to f at that point. By precomposing it with the diagonal map Δ, x → (x, x), a generalized Taylor series may be begun as

${\displaystyle {\begin{aligned}f(\mathbf {x} )&\approx f(\mathbf {a} )+(Df)(\mathbf {x-a} )+\left(D^{2}f\right)(\Delta (\mathbf {x-a} ))+\cdots \\&=f(\mathbf {a} )+(Df)(\mathbf {x-a} )+\left(D^{2}f\right)(\mathbf {x-a} ,\mathbf {x-a} )+\cdots \\&=f(\mathbf {a} )+\sum _{i}(Df)_{i}(x_{i}-a_{i})+\sum _{j,k}\left(D^{2}f\right)_{jk}(x_{j}-a_{j})(x_{k}-a_{k})+\cdots \end{aligned}}}$

where f(a) is identified with a constant function, x_i − a_i are the components of the vector x − a, and (Df)_i and (D²f)_jk are the components of Df and D²f as linear transformations.