# Computing Derivatives

### Topics

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- Check the answer.

### Sample Problem

Find the derivative of the function *h*(*x*) = cos(sin(*ln* *x*))).

Answer.

- Figure out what the problem is asking.

The problem is asking us to find a derivative. More specifically, it's asking us to use the chain rule to find a derivative. We can tell this because there are nested functions. In fact, the functions are nested 3 deep. We will use the chain rule 2 times to find the derivative we want.

- Solve the problem.

cos(□) is the outside function and {sin(*ln**x*)} is the inside function. Then the chain rule says

*h'*(*x*) = -sin({sin(*ln**x*)}) × ({sin (*ln**x*)})'.

We need to use the chain rule again to find (sin (*ln* x))'. The outside function is now sin(□) and the inside function is *ln* x, therefore

Now we can go back to our first application of the chain rule and simplify:

- Check the answer.

There's no great way to check this answer. We could do it over again and make sure we find the same answer the second time; we could check the answer in the book; we could have a calculator or computer compute the derivative and compare it to our answer.

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