## Theorem 6.5.3: Derivative as Linear Approximation

Let

*f*be a function defined on*(a, b)*and*c*any number in*(a, b)*. Then*f*is differentiable at*c*if and only if there exists a constant*M*such thatwhere the remainder functionf(x) = f(c) + M ( x - c ) + r(x)

*r(x)*satisfies the condition= 0

### Proof:

First, suppose f is differentiable at x = c. Let the constant M = f'(c) and set

- r(x) = f(x) - f(c) - f'(c) ( x - c )

We have to check the limit of the quotient

- = - f'(c)

Since f is differentiable, the limit of this expression is zero as x approaches c, as required.

Second, suppose that f(x) = f(c) + M ( x - c ) + r(x) for some constant M and = 0 Then

- - M =

The limit on the right as x approaches c is zero by assumption. Hence, the limit on the left must also be zero, and we recognize the constant M as the derivative of the function f'(c).