1. 2.5 The Chain Rule If f and g are both differentiable and F is the composite function defined by F(x)=f(g(x)), then F is differentiable and F′ is given by the product In Leibniz notation, if y=f(u) and u=g(x) are both differentiable functions, then

2. Example: A faster way to write the solution: Differentiate the outer function... …then the inner function

3. Another example: derivative of the outer function derivative of the inner function It looks like we need to use the chain rule again!

4. Another example: The chain rule can be used more than once. (That’s what makes the “chain” in the “chain rule”!)

5. The Power Rule combined with the Chain Rule If n is any real number and u=g(x) is differentiable, then Alternatively, Example:

7. 2.6 Implicit Differentiation The functions we considered so far can be described by expressing one variable explicitly in terms of another variable, e.g. y = x 2 + 3x or y = cos x However, some functions are defined implicitly by a relation between x and y such as x 2 + y 2 = 1 or 2y = x 2 + sin y For most of this kind of functions, it’s not easy to solve for y to express it explicitly as a function of x. But fortunately we don’t need to solve an equation for y in order to find the derivative of y. Instead we can use the method of implicit differentiation: 1 Differentiate both sides w.r.t. x. 2 Solve for.

8. 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. Example 1:

9. This can’t be solved for y. Example 2:

10. We need the slope. Since we can’t solve for y, we use implicit differentiation to solve for. Find the equation of the tangent line to the curve at (-1, 2) tangent line: slope: Example 3:

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