Partial Derivatives

Partial derivatives are a way to derive functions that have more than one independent variable. They have a wide range of uses, including topics in Physics, Calculus, Economics, and Computer Science. This handout will focus on the fundamental techniques for solving differential equations using partial derivatives.

Notation

Partial derivatives use a notation that is intentionally similar to that of regular derivatives. Their overall format is the same, but the shorthand symbols, such as “dx,” are replaced by the

stylized version of the letter d: ∂. Partial derivatives should be labeled as or , depending on ∂ ∂ the variable being derived. The two derivatives should be read as “The partial∂x de∂yrivative, with respect to x,” and “The partial derivative, with respect to y.”

Process for Solving

Partial derivatives follow the same procedures and rules of differentiation as normal derivatives with one exception. When taking a partial derivative with respect to a particular variable, treat all other variables as though they are constants such as when deriving with respect to x, treat y as a constant, and do not derive the y. When deriving with respect to , do not derive . This

method may at first seem similar to implicit differentiation; however, 𝑦𝑦there is a crucial 𝑥𝑥 difference between the two methods. Implicit differentiation can be used when depends on

the independent variable. When each variable is independent of the other, the𝑦𝑦 partial derivative method𝑥𝑥 should be used.

This handout will only cover using the Constant, Product, and Power Rules of differentiation for solving a partial derivative. However, a complete list of the rules can be found in any Calculus textbook or in our Common Derivatives and Integrals handout, which is also located on the

Academic Center for Excellence website at www.germanna.edu/academic-center-for- excellence/helpful-handouts.

The mathematical definitions of these three rules of differentiation are as follows: Constant Rule: ( ) = 0 ′ Product Rule: 𝑎𝑎( ) ( ) = ( ) ( ) + ( ) ( ) ′ ′ Power Rule: �𝑓𝑓( 𝑥𝑥) ∙ 𝑔𝑔 =𝑥𝑥 � 𝑓𝑓 𝑥𝑥 ∙ 𝑔𝑔 𝑥𝑥 𝑓𝑓 𝑥𝑥 ′ ∙ 𝑔𝑔 𝑥𝑥 𝑏𝑏 ′ 𝑏𝑏−1 𝑎𝑎𝑥𝑥 𝑎𝑎𝑎𝑎𝑥𝑥 Example Problem Take the partial derivatives of the following function with respect to and then :

𝑥𝑥 𝑦𝑦 ( , ) = + 2 2 3 𝑓𝑓 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦 𝑥𝑥 To take the partial derivative of this function with respect to , apply the Product, Power, and

Constant Rules to the term, and apply the Power Rule to𝑥𝑥 the 2 term. 2 3 𝑥𝑥 𝑦𝑦 𝑥𝑥 To apply the Product Rule to the term, let ( ) represent and ( ) represent . First, 2 2 derive ( ) by using the Power Rule𝑥𝑥 𝑦𝑦 to yield a derivative𝑓𝑓 𝑥𝑥 of 2 𝑥𝑥. Secondly,𝑔𝑔 𝑥𝑥 derive ( ).𝑦𝑦 Because𝑓𝑓 𝑥𝑥( ) represents , and the partial derivative being taken𝑥𝑥 is with respect𝑔𝑔 to𝑥𝑥 , ( ) will be considered𝑔𝑔 𝑥𝑥 a constant 𝑦𝑦per the Constant Rule and yield a derivative of 0. Combining𝑥𝑥 𝑔𝑔the𝑥𝑥 ( ) and ( ) pieces of the Product Rule will yield the following: 𝑓𝑓 𝑥𝑥 𝑔𝑔 𝑥𝑥

( ) = 2 + 0 = 2 𝜕𝜕𝑓𝑓 2 2 Applying the𝑥𝑥 Power𝑦𝑦 R𝑥𝑥𝑦𝑦ule to𝑥𝑥 the∙ equation’s𝑥𝑥𝑦𝑦 second term will yield 6 . After combining both 𝜕𝜕𝑥𝑥 2 parts, the final result of the partial derivative with respect to is: 𝑥𝑥 𝑥𝑥 = 2 + 6 𝜕𝜕𝑓𝑓 2 𝑥𝑥𝑦𝑦 𝑥𝑥 𝜕𝜕𝑥𝑥 Provided by Academic Center for Excellence 1 Partial Derivatives September 2018

When taking the partial derivative with respect to , the Power, Product, and Constant Rules are used to find the derivative of the first term, and𝑦𝑦 the Constant Rule alone is applied to find the derivative of the second term.

For the first term, the and still correspond to ( ) and ( ) respectively; however, their 2 derivatives have changed𝑥𝑥 . The𝑦𝑦 Constant Rule must𝑓𝑓 be𝑥𝑥 used t𝑔𝑔o derive𝑥𝑥 because variables other 2 than are now treated as constants. Use the Power Rule to take the 𝑥𝑥derivative of , which will result𝑦𝑦 in a value of 1. Finally, the two halves of the Product Rule can be reassembled𝑦𝑦 .

( ) = 0 + 1 = 𝜕𝜕𝑓𝑓 2 2 2 𝑥𝑥 𝑦𝑦 ∙ 𝑦𝑦 𝑥𝑥 ∙ 𝑥𝑥 𝜕𝜕𝑦𝑦 When deriving the second term, only the Constant Rule is needed since 2 does not contain 3 any variables. The derivative will, therefore, be 0. Combining the terms 𝑥𝑥together will give the final𝑦𝑦 result for the partial derivative with respect to . 𝑦𝑦 = + 0 = 𝜕𝜕𝑓𝑓 2 2 𝑥𝑥 𝑥𝑥 𝜕𝜕𝑦𝑦 This handout specifically illustrates how to use the Constant, Power, and Product Rules when taking a partial derivative, but the same process can also be applied to any of the other rules of differentiation. For any problem, identify the variable that needs to be derived, then treat all other variables as constants and apply any derivative rules that are necessary to solve the equation.

Provided by Academic Center for Excellence 2 Partial Derivatives September 2018

Practice:

Find the partial derivatives with respect to each variable present:

1. ( , ) = 3 2 𝑓𝑓 𝑥𝑥 𝑦𝑦 𝑥𝑥 𝑦𝑦

2 2 2. ( , ,V) = 2 + + 𝑉𝑉 𝑉𝑉 𝑆𝑆 𝑙𝑙 𝑤𝑤 𝑙𝑙𝑤𝑤 𝑙𝑙 𝑤𝑤

3. ( , ) = 2 2𝑥𝑥 −4 𝑙𝑙 𝑥𝑥 𝑇𝑇 𝑒𝑒 𝑇𝑇

4. ( , , ) = + 7 2 −1 𝑝𝑝 𝑇𝑇 𝑣𝑣 ℎ 𝑇𝑇 ℎ𝑣𝑣 ℎ𝑣𝑣 −𝑇𝑇

5. ( , , ) = sin (3 ) 2𝑦𝑦 4 𝑓𝑓 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑥𝑥 𝑒𝑒 𝑧𝑧

Provided by Academic Center for Excellence 3 Partial Derivatives September 2018

Answers to the Above Problems

1. = 6x , = 3 𝜕𝜕𝑓𝑓 𝜕𝜕𝑓𝑓 2 𝑦𝑦 𝑥𝑥 𝜕𝜕𝑥𝑥 2𝜕𝜕𝑦𝑦 2 2 2 2. = 2 , = 2 , = + 𝜕𝜕𝑆𝑆 𝑉𝑉 𝜕𝜕𝑆𝑆 𝑉𝑉 𝜕𝜕𝑆𝑆 𝑤𝑤 − 2 𝑙𝑙 − 2 𝜕𝜕𝑙𝑙 𝑙𝑙 𝜕𝜕𝑤𝑤 𝑤𝑤 𝜕𝜕𝑉𝑉 𝑙𝑙 𝑤𝑤 3. = 4 , = 8 𝜕𝜕𝑙𝑙 2𝑥𝑥 −4 𝜕𝜕𝑙𝑙 2𝑥𝑥 −5 𝑒𝑒 𝑇𝑇 − 𝑒𝑒 𝑇𝑇 𝜕𝜕𝑥𝑥 𝜕𝜕𝑇𝑇 4. = 2 1, = + , = + 7 𝜕𝜕𝑝𝑝 −1 𝜕𝜕𝑝𝑝 2 −2 𝜕𝜕𝑝𝑝 2 −1 𝑇𝑇ℎ𝑣𝑣 − −𝑇𝑇 ℎ𝑣𝑣 7ℎ 𝑇𝑇 𝑣𝑣 𝑣𝑣 𝜕𝜕𝑇𝑇 𝜕𝜕𝑣𝑣 𝜕𝜕ℎ 5. = 3cos(3 ) , = 2sin (3 ) , = 4sin (3 ) 𝜕𝜕𝑓𝑓 2𝑦𝑦 4 𝜕𝜕𝑓𝑓 2𝑦𝑦 4 𝜕𝜕𝑓𝑓 2𝑦𝑦 3 𝑥𝑥 𝑒𝑒 𝑧𝑧 𝑥𝑥 𝑒𝑒 𝑧𝑧 𝑥𝑥 𝑒𝑒 𝑧𝑧 𝜕𝜕𝑥𝑥 𝜕𝜕𝑦𝑦 𝜕𝜕𝑧𝑧

Provided by Academic Center for Excellence 4 Partial Derivatives September 2018