Partial Derivative: Definition, Rules & Examples

Partial Derivative: Definition, Rules & Examples
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  • 0:04 Partial Derivative Definition
  • 3:04 Second Order Derivatives
  • 7:06 Extending Partial Derivatives
  • 8:23 Other Derivative Notations
  • 9:00 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

When a function depends on more than one variable, we can use the partial derivative to determine how that function changes with respect to one variable at a time. In this lesson, we use examples to define partial derivatives and to explain the rules for evaluating them.

Partial Derivative Definition

Calories consumed and calories burned have an impact on our weight. Let's say that our weight, u, depended on the calories from food eaten, x, and the amount of physical exertion we do, y. If we only regulated our eating, while doing the same exercise every day, we could ask how does u change when we vary only x. Likewise, we could keep x constant and take note of how u varies when we change y. This would be like keeping a constant daily diet while changing how much we exercise. This idea of change with respect to one variable while keeping other variables constant is at the heart of the partial derivative.

When we write u = u(x,y), we are saying that we have a function, u, which depends on two independent variables: x and y. We can consider the change in u with respect to either of these two independent variables by using the partial derivative. The partial derivative of u with respect to x is written as:


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What this means is to take the usual derivative, but only x will be the variable. All other variables will be treated as constants. We can also determine how u changes with y when x is held constant. This is the partial of u with respect to y. It's written as:


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For example, if u = y * x^2, then:


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Likewise, we can differentiate with respect to y and treat x as a constant with the equation:


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The rule for partial derivatives is that we differentiate with respect to one variable while keeping all the other variables constant. As another example, find the partial derivatives of u with respect to x and with respect to y for:


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To do this example, we will need the derivative of an exponential with the following:


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And the derivative of a cosine, which is written as:


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Thus:


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and


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Second Order Derivatives

So far we have defined and given examples for first-order partial derivatives. Second-order partial derivatives are simply the partial derivative of a first-order partial derivative. We can have four second-order partial derivatives, which you can see right here:


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Continuing with our first example of u = y * x^2,


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and


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Likewise,


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and


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And in our second example:


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and


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The mixed partial derivatives become:


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and


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