Derivatives

Classes: Partial Derivative Review and Applied Problems
Examples are taken from Thomas’s Calculus textbook 10th Ed.



Review: Partial Derivative Example


What is apartial derivative?
A partial derivative with respect to[pic]at the point [pic] is the ordinary derivative of[pic] with respect to[pic]at the point[pic].


When are they used?
Apartial derivative is used for functions with more than one independent variable.


How do you calculate it?
The partial derivative is taken exactly like a normal derivative, with oneexception. All but one independent variable are treated like a constant. If you take the derivative with respect to x, then y is treated as a constant.


Example:
Let[pic]. Find thevalues of [pic]and [pic]at the point (4, -5).
SOLUTION:
To find [pic], we treat y as a constant and differentiate with respect to x.
[pic]


The value of [pic] at (4,-5) is 2(4) + 3(-5) = -7.
To find [pic], we treat x as a constant and differentiate with respect to y:

[pic].

The value of [pic]at (4, -5) is 3(4) + 1 = 13.

See chapter 11section 3 of Thomas’s Calculus 10th Ed. for more examples.
















Uses: There are many uses of a partial derivative.

First you can use it for gradient. Whatis a gradient?

A gradient is a vector obtained by evaluating the partial derivatives of a function at P0. Once it is calculated from the partial derivatives, it can be used to calculate adirectional derivative. This directional derivative then tells rate of change of a function in a particular direction.

The gradient of[pic]in Cartesian coordinates:[pic]

Gradientexample:

Find the derivative of f(x,y)=xey + cos(xy) at the point (2,0) in the direction of v=3i-4j.

First, calculate the gradient by taking the two partial derivatives.

fx=ey – y...