6.5. Differentiable Functions
Theorem 6.5.3: Derivative as Linear Approximation
f(x) = f(c) + M ( x - c ) + r(x)where the remainder function r(x) satisfies the condition
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).