Derivatives and their applications

Common derivative functions

From f'(a) to the function f'

When f is differentiable at every point in an interval, we can define a new function, the derivative function f', which associates the derivative f'(x) with each x. To avoid having to recalculate the limit each time, we use formulas that we know off by heart.

Derivatives of standard functions

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

Operations on derivatives

Let u and v be two differentiable functions, and k a real constant:

(u + v)' = u' + v'
(k * u)' = k * u'
(u * v)' = u' * v + u * v'
(u/v)' = (u' * v - u * v') / v² (if v is non-zero)

Practical example

Let’s differentiate f(x) = 3x² + 5x - 7:

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

Another example involving a product: g(x) = x² * (x + 1). Let u = x² (u' = 2x) and v = x + 1 (v' = 1):

g'(x) = 2x * (x+1) + x^2 * 1 = 2x^2 + 2x + x^2 = 3x^2 + 2x

Common pitfall