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
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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