- Topics At a Glance
- Derivatives of Basic Functions
- Constant Functions
- Lines
- Power Functions
- Exponential Functions
- Logarithmic Functions
- Trigonometric Functions
- Derivatives of More Complicated Functions
- Derivative of a Constant Multiple of a Function
- Multiplication by -1
- Fractions With a Constant Denominator
- More Derivatives of Logarithms
- Derivative of a Sum (or Difference) of Functions
- Derivative of a Product of Functions
- Derivative of a Quotient of Functions
- Derivatives of Those Other Trig Functions
- Solving Derivatives
- Using the Correct Rule(s)
- Giving the Correct Answers
- Derivatives of Even More Complicated Functions
- The Chain Rule
- Re-Constructing the Quotient Rule
- Derivative of a
^{x} - Derivative of
*ln*x - Derivatives of Inverse Trigonometric Functions
- The Chain Rule in Leibniz Notation
- Patterns
- Thinking Backwards
- Implicit Differentiation
- Computing Derivatives Using Implicit Differentiation
- Using Leibniz Notation
- Using Lagrange Notation
**In the Real World**- I Like Abstract Stuff; Why Should I Care?
**How to Solve a Math Problem**

There are three steps to solving a math problem.

- Figure out what the problem is asking.

- Solve the problem.

- Check the answer.

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.