Derivatives and graphical analysis

The derivative and common formulas

From f'(a) to the derivative function

When f is differentiable at every point a in an interval I, we can associate the number f'(x) with each real number x. We thus define a new function, the derivative function of f, denoted by f'.

Rather than recalculating the limit each time, we use well-known differentiation formulas, which must be learnt by heart.

Table of common derivatives

Function f(x)	Derivative f'(x)	Domain of validity
k (constant)	0	for all x
x	1	for all x
x²	2x	for all x
x^n (n an integer ≥ 1)	n * x^(n-1)	for all x
1/x	-1/x²	x ≠ 0
√x	1/(2√x)	x > 0

Rules of differentiation

For two differentiable functions u and v and a real number k:

(u + v)' = u' + v' (k * u)' = k * u' (u * v)' = u' * v + u * v'

Example

Let f(x) = 3x² - 5x + 2. Using the rules above:

f'(x) = 3 * (2x) - 5 * 1 + 0 = 6x - 5

Thus f'(1) = 6 × 1 − 5 = 1, which directly gives the slope of the tangent at x = 1 without having to use the limit.

Common pitfall