1. 3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

2. This can’t be solved for y. Since it can’t, the methods of differentiation that we’ve learned so far won’t work. In those cases, y was defined as an explicit function of x y = f (x) Since y can’t be explicitly defined here, the methods of differentiation that we’ve learned so far won’t work. In the case above, since we can’t solve explicitly for y, it is considered an implicit function.

3. We know that… DON’T FORGET THE BABY! …and that… …but what about…

4. An extension of the Chain Rule: Usually, the inside function in a chain rule problem is a function we know like sin x or x 2. In this case, consider y to be a function of x that we don’t know so we just give it the generic derivative symbol This extention of the Chain Rule is called implicit differentiation

5. An extension of the Chain Rule: Usually, the inside function in a chain rule problem is a function we know like sin x or x 2. In this case, consider y to be a function of x that we don’t know so we just give it the generic derivative symbol Where

6. An extension of the Chain Rule: Where But with implicit differentiation… But we already know that So…

7. Since y is just the “inside” function of the chain rule problem… is just the “baby” derivative

8. This is not a function, but we can still find the slope of any tangent lines. Do the same thing to both sides. Note use of chain rule.

9. Remember that this can’t be solved for y. Implicit differentiation 1 Differentiate both sides w.r.t. x. 2 Solve for.

10. 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.

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

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