Definition: The derivative of a function f at a number a , denoted by f'(a), is given by

f'(a) = lim_h->0 [f(a+h)-f(a)]/h
if this limit exists!

The derivative is therefore the same as the slope of the tangent line to f(x) at x=a if it exists.

An equation of the tangent line to the curve y=f(x) at the point x=a is given by

y - f(a) = f'(a)*(x-a) if f'(a) exists.

Often, if an explicit limit cannot be computed, a numerical approximation is useful, as when f(x)=2^x. What is f'(0)?

If y=f(x) is smooth at x=a, and one "zooms" in to the point (a,f(a)), then the slope of the straight line will approach m=f'(a).

If we let y₁ = f(x₁) and y₂ = f(x₂), and delta_y = y₂-y₁, and delta_x = x₂-x₁, then the derivative is approximately equal to

delta_y / delta_x and in the limit f'(x₁) = lim _{delta_x -> 0} delta_y/delta_x

Definition: The derivative of a function f at x , denoted by f'(x), is given by

f'(x) = lim_h->0 [f(x+h)-f(x)]/h
if this limit exists!

Consequently, f'(x) is itself a function .

This process may be repeated as long as the appropriate limits exist.

Theorem: If f is differentiable at a, then f is continuous at a.

Thus differentiability implies continuity!

A function can fail to be differentiable at x=a if one or more of the following occurs:

• The left and right limits do not agree ("corner")
• The limit does not exist ("cusp")
• f(x) is not continous at x=a

Last modified Wed Sep 18 20:59:23 CDT 1996