1. Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

2. Explicit and Implicit Functions Explicit functions can be written in the form We have differentiated many explicit functions so far! Implicit functions cannot easily be solved for And sometimes they aren’t even functions!

3. This is not a function, but it would still be nice to be able to find the slope of the tangent. Do the same thing to both sides. Consider the circle Note use of chain rule. This is because we are differentiating with respect to

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

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

6. 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.

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

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