Slide 3- 1 Quick Quiz Sections 3.4 – 3.6

3.7 Implicit Differentiation

Slide 3- 3 What you’ll learn about Implicitly Defined Functions Lenses, Tangents, and Normal Lines Derivatives of Higher Order Rational Powers of Differentiable Functions … and why Implicit differentiation allows us to find derivatives of functions that are not defined or written explicitly as a function of a single variable.

Slide 3- 4 Implicitly Defined Functions

Slide 3- 5 Implicitly Defined Functions

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.

Slide 3- 7 Example Implicitly Defined Functions

Slide 3- 8 Implicit Differentiation Process

This can’t be solved for y. This technique is called implicit differentiation. 1 Differentiate both sides w.r.t. x. 2 Solve for.

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

Find the equations of the lines tangent and normal to the curve at. tangent:normal:

Slide Rule 9 Power Rule For Rational Powers of x

Slide Example Derivatives of a Higher Order

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