Derivatives

Derivative of a function f at a number a

Notation

We denote the derivative of a function f at a number a as f'(a)\,\!.

Definition

The derivative of a function f at a number a a is given by the following limit (if it exists):

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


An analagous equation can be defined by letting x=(a+h). Then h=(x-a), which shows that when x approaches a, h approaches 0:

f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}

Interpretations

As the slope of a tangent line

Given a function y=f(x)\,\!, the derivative f '(a)\,\! can be understood as the slope of the tangent line to f(x) at x=a:

As a rate of change

The derivative of a function f(x) at a number a can be understood as the instantaneous rate of change of f(x) when x=a.

At a tangent to one point of a curve

Vocabulary

The point A(a ; f(a)) is the point in contact of the tangent and Cf.

Definition

If f is differentiable in a, then the curve C admits at a point A which has for coordinates (a ; f(a)), a tangent : it is the straight line passing by A and of direction coefficient f'(a). An equation of that tangent is written: y = f'(a)*(x-a)+f(a)


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