Implicit Differentiation Examples

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Implicit Differentiation Examples

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Implicit Differentiation Examples Youtube

In implicit differentiation we differentiate each side of an equation with two variables usually x and y by treating one of the variables as a function of the other This calls for using the chain rule Let s differentiate x 2 y 2 1 for example Here we treat y as an implicit function of x Example: x 2 + y 2 = r 2. Differentiate with respect to x: d dx (x 2) + d dx (y 2) = d dx (r 2) Let's solve each term: Use the Power Rule: d dx (x2) = 2x. Use the Chain Rule (explained below): d dx (y2) = 2y dy dx. r 2 is a constant, so its derivative is 0: d dx (r2) = 0. Which gives us: 2x + 2y dy dx = 0. Collect all the dy dx on one side.

Implicit Differentiation Examples Youtube

Implicit Differentiation ExamplesFor example, x²+y²=1. Implicit differentiation helps us find dy/dx even for relationships like that. This is done using the chain rule, and viewing y as an implicit function of x. For example, according to the chain rule, the derivative of y². Example 70 Using Implicit Differentiation Given the implicitly defined function sin x 2y 2 y 3 x y find y prime Solution Differentiating term by term we find the most difficulty in the first term It requires both the Chain and Product Rules

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