*Tiffany Price*Show bio

Tiffany has many years of classroom teaching experience and has a masters degree in Educational Leadership.

Lesson Transcript

Instructor:
*Tiffany Price*
Show bio

Tiffany has many years of classroom teaching experience and has a masters degree in Educational Leadership.

In this lesson, we'll introduce partial derivative equations and look into how they are solved. We'll explore several of the derivative rules and apply those rules to solve a few examples.

**Derivatives**, in general, are important in calculus in that they allow us to see the value of a function at a particular point. Partial derivatives are very similar to solving total derivatives because the same rules apply to both. Total derivatives allow all of the variables in an equation to change, while **partial derivatives** only differentiate one variable at a time while the other variable(s) remain fixed.

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The easiest way to solve both partial and total derivatives is to memorize the shortcut derivative rules or have a chart of the rules handy. A few of the rules for solving derivative equations are:

- The
**constant rule**: the derivative of a constant is 0

- The
**line rule**: the derivative of*nx*(a first order variable) is equal to its coefficient (*n*)

- The
**power rule**: the derivative of*x**n*(any variable to a power) is*nx*(*n*-1)

**Logarithms rule**: the derivative of ln(*x*) is 1/*x*

**Chain rule**: the derivative of more complicated functions is (derivative of the outside function leaving inside alone) x (derivative of inside function)

There are many more rules than these, but in this lesson, we will only look at these five. When solving partial derivatives, the variable that is not being differentiated is treated like a constant.

Also, be aware that there is no uniform symbol to represent differentiation. Here are a few notations you may come across when solving these functions:

Let's solve a couple of equations:

Find the derivative of this equation with respect to x: *f(x,y) = 6x - 9y³*

df/dx (*x*,*y*) = 6*x* - 9*y³*

Because the derivative of 6*x* is 6, and - 9*y³* is treated like a constant (whose derivative is 0), the answer is 6

df/dx (*x,y*) = 6 - 0 = 6

Let's look at the same equation and differentiate the y variable only.

df/dy (*x,y*) = 6*x* - 9*y³*

This time the derivative of 6*x* will be 0 because we're treating it as a constant. The derivative of -9*y³* is - 27*y²* using the power rule.

df/dy (*x,y*) = 0 - 9(3)y(3-1) = - 27*y²*

Now, we're going to differentiate a more complicated function and use the chain rule and logarithm rule. Don't let the z at the beginning of the function throw you off. When using the chain rule to differentiate multivariable functions, you must have a variable to tie all the other variables together. In this case, that variable is z. Luckily for us, in order to differentiate this function, the z doesn't come into play. Let's take a look at how to differentiate the function with respect to x, then we will see how to differentiate it with respect to y.

*z* = f(*x,y*) = (*x²* + *y³*)9 + ln(*x*)

Let's take a few moments to review the most important information that we learned in this lesson on partial derivatives. **Derivatives**, in general, allow us to assess the change within a function to solve a problem at a particular point. **Partial derivatives** only allow us to alter one variable at a time while holding the other(s) constant, but the concept and the rules are very similar to those of total derivatives.

We also looked at the following 5 (of the many that exist) derivative rules in this lesson:

- The
**constant rule**: the derivative of a constant is 0

- The
**line rule**: the derivative of*nx*(a first order variable) is equal to its coefficient (*n*)

- The
**power rule**: the derivative of*x**n*(any variable to a power) is*nx*(*n*-1)

**Logarithms rule**: the derivative of ln(*x*) is 1/*x*, and

**Chain rule**: the derivative of more complicated functions is (derivative of the outside function leaving inside alone) x (derivative of inside function)

Using these different derivative rules should allow you to solve most partial derivative equations that come your way.

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