Section 3
Partial Derivatives
By Boundless
![Thumbnail](../../../../../../figures.boundless-cdn.com/17883/square/surface-plot.jpg)
Multivariable calculus is the extension of calculus in one variable to calculus in more than one variable.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17882/square/ple-of-continuous-function.jpg)
A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17881/square/x2-2bx-2b1.jpg)
A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17886/raw/image-tangent-plane.jpg)
The tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17887/square/scalar-field.jpg)
For a function
![Thumbnail](../../../../../../figures.boundless-cdn.com/17888/square/gradient99.jpg)
The directional derivative represents the instantaneous rate of change of the function, moving through
![Thumbnail](../../../../../../figures.boundless-cdn.com/17889/square/saddle-point.jpg)
The second partial derivative test is a method used to determine whether a critical point is a local minimum, maximum, or saddle point.
![Thumbnail](../../../../../../figures.boundless-cdn.com/29391/square/nontangential.jpg)
The method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints.
![Thumbnail](../../../../../../figures.boundless-cdn.com/17893/raw/cuboid-simple.jpg)
To solve an optimization problem, formulate the function
![Thumbnail](../../../../../../figures.boundless-cdn.com/17894/square/fig1.jpeg)
Finding extrema can be a challenge with regard to multivariable functions, requiring careful calculation.