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

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 = x2 + 3x or y = cos x
However, some functions are defined implicitly by a relation between x and y such as
x2 + y2 = 1 or 2y = x2 + 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. Example 1:
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.

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

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

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