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Introduction to Computing Derivatives - At A Glance:

There are three steps to solving a math problem.

  1. Figure out what the problem is asking.
      
  2. Solve the problem.
      
  3. 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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