Calculus: IMPLICIT DIFFERENTIATION Section 4.5. Explicit vs. Implicit y is written explicitly as a function of x when y is isolated on one side of the.

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Presentation transcript:

Calculus: IMPLICIT DIFFERENTIATION Section 4.5

Explicit vs. Implicit y is written explicitly as a function of x when y is isolated on one side of the equation. When y is not isolated we say the function is written implicitly.

Why do we need implicit differentiation? Example: Try to solve x 2 y + y 2 x = -2 for y. This is not possible. So in order to take the derivative, we need this new technique …

This is not a function, but it would still be nice to be able to find the slope. Do the same thing to both sides. Note use of chain rule.

Keep in Mind! When you differentiate terms involving y, you must apply the Chain Rule, because you are assuming that y is defined implicitly as a function of x.

Guidelines for Implicit Differentiation Differentiate both sides of the equation with respect to x. Collect all terms involving dy/dx on the left side of the equation and move all other terms to the right side of the equation. Factor dy/dx out of the left side of the equation. Isolate dy/dx by dividing.

Board Examples x 2 y + y 2 x = -2

We need the slope. Since we can’t solve for y, we use implicit differentiation to solve for. Find the equations of the line tangent to the curve at. Note product rule.

tangent: Find the equations of the line tangent to the curve at.

Higher Order Derivatives Find if. Substitute back into the equation.

Assignment Page 297: 1-31 odds, 35, 39