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Implicit Differentiation: Definition, Examples

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Derivatives > Implicit Differentiation

implicit differentiation

Implicit differentiation is used when it’s difficult, or impossible to solve an equation for x. For example, the functions y=x2/y or 2xy = 1 can be easily solved for x, while a more complicated function, like 2y2 -cos y = x2 cannot. When you have a function that you can’t solve for x, you can still differentiate using implicit differentiation. With this technique, you directly differentiate both sides of the equation without solving for x.

Implicit Differentiation

Example problem #1: Differentiate 2x-y = -3 using implicit differentiation.

Step 1: Write out the function with the derivative on both sides:
dy/dx [2x-y] = dy/dx [-3]
This step isn’t technically necessary but it will help you keep your calculations tidy and your thoughts in order.

Step 2: Differentiate the right side of the equation. The right side of this equation is a constant, so the derivative is zero:
dy/dx [2x-y] = 0

Step 2: Differentiate the left side of the equation. The derivative of 2x-y is 2 (using the power rule and constant rule). Remember to treat the dependent variable as a function of the dependent variable:
2- dy/dx = 0

Step 4: Use algebra to solve for the derivative.
dy/dx = 2.

That’s it!


Example problem #2: Differentiate y2 + x2 = 7 using implicit differentiation.

Step 1: Differentiate the left and right sides of the equation. This example also uses the power rule and constant rule:
2y dy/dx + 2x = 0


Step 2: Use algebra to solve:
2y dy/dx + 2x = 0
2y dy/dx = -2x
dy/dx = -2x/2y
dy/dx = -x/y

That’s it!

Tip: These basic examples show how to perform implicit differentiation using the power rule and constant rule. Depending on what function you are trying to differentiate, you may need to use other techniques of differentiation, including the chain rule, to solve.
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Stephanie Glen. "Implicit Differentiation: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/implicit-differentiation/
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