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Using the Definition to Compute the Derivative - Exercise 4

**Exercise 4.** Show that if *f*'(*a*) exists, then we have

**Answer.** We know that

Equivalently, we also have

Putting these two equations together we will get

But

which gives the desired relation.
Note that this formula is for example used by calculators to
approximate *f*'(*a*) especially when *f*(*x*) is known for values of
*x* near *a*.

In the same spirit as the example above, one can prove the
following formula which involves the second derivative:

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