3.7 Implicit Differentiation

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

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 xy = 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… …and that… …but what about… DON’T FORGET THE BABY!

4. An extension of the Chain Rule: This extention of the Chain Rule is called implicit differentiation Usually, the inside function in a chain rule problem is a function we know like sin x or x2. 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

5. An extension of the Chain Rule: Where Usually, the inside function in a chain rule problem is a function we know like sin x or x2. 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

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

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

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

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