# The Chain Rule

The Chain Rule computes the derivative of the composite of two functions. The composite is just "g inside f" --- that is,

(Note that this is not multiplication!)

Here are some examples:

Here's a more complicated example:

One way to tell which function is "inside" and which is "outside" is to think about how you would plug numbers in. For example, take . What would you do to compute on your calculator? First, you'd square 1.7 --- . Next, you'd take the sine of that --- .

The function you did first --- squaring --- is the inner function. The function you did second --- sine --- is the outer function.

Example. Suppose

Compute , , and .

The Chain Rule says that

In words, you differentiate the outer function while holding the inner function fixed, then you differentiate the inner function.

Example. Compute .

looks like this:

Differentiate the outer function , obtaining . What is "junk"? It's . The first term in the Chain Rule is . (Notice that I differentiated the outer function, temporarily leaving the inner one untouched.)

Next, differentiate the inner function. The derivative of is .

Therefore,

Example. Compute .

While it would be correct to use the Quotient Rule, it's unnecessary.

In general, you do not need to use the Quotient Rule to differentiate things of the form

In the first case, use the Chain Rule as above. In the second case, divide the top by the number on the bottom.

Example. Compute .

Example. Compute .

Recall the derivative formula for sine:

Example. Compute .

Recall the derivative formula for cosine:

Therefore,

Example. f and g are differentiable functions. A table of some values for these functions is shown below.

Find .

By the Chain Rule,

Example. Compute .

Example. Notice that

Do you understand the difference between and ? Here's a picture:

In the first case, the outer function is the squaring function; in the second case, the outer function is the sine function.

Example. Recall that

So

Example. Compute .

Differentiate from the outside in:

Example. Where does the graph of have a horizontal tangent?

Set and solve for x:

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