The Definition of the Derivative

The derivative at a point

The derivative $$f'(x)$$ of a function $$f(x)$$ at a point $$(a, f(a))$$ is defined by

$f'(a)=\lim_{h\rightarrow 0}\frac{f(a+h)-f(h)}{h}$

The value of

$\frac{f(a+h)-f(h)}{h}$

is the slope of the line through the points $$(a, f(a))$$ and $$(a+h, f(a+h))$$, the so called secant line.

Note that $$\Delta x = a+h-a=h$$ and $$\Delta y = f(a+h)-f(a)$$. The limit of the secant lines as $$h$$ tends to zero is the tangent line. The derivative is the slope of the tangent line to the graph at the point where $$x=a$$. Finding the derivative is called differentiating.

What does the graph look like when the derivative is positive, negative, zero?

Taking the absolute value of the function above, you get the graph below.

Say something interesting about the derivative of the function above!

The derivative as a function

You can extend the definition of the derivative at a point to a definition concerning all points (all points where the derivative is defined, i.e. where the limit exists); if doing so you get a new function $$f'(x)$$ defined like this:

$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Finding an algebraic formula for the derivative of a function by using the definition above, is sometimes called differentiating from first principle.

By using a computer you can find numerical approximations of the derivative at all points of the graph.

The line shown in the construction below is the tangent to the graph at the point A. The slope of the tangent line is the y-value of the point P. Drag the point A to see the trace of the point P, this trace is the graph of the derivative function.

Note that the point P does not move when changing the slider e, hence there are infinitely many functions giving rise to the same derivative, these functions differ by a constant but their graphs have the "same shape".

Another way of writing it

If you instead use $$y$$ to denote a function of a variable $$x$$, $$y=f(x)$$, then you use the notation $$\frac{dy}{dx}$$ to denote the derivative. The notation stems from the definition

$\frac{dy}{dx}=\lim_{\Delta x \rightarrow 0}\frac{\Delta y}{\Delta x}$