Calculus: Derivative Rules

f'(x) = \displaystyle\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}

The instantaneous rate of change of f, which is the slope of the tangent line to the graph at that point.

n x^{n-1}

0 and c f'(x)

f'g + fg'

\dfrac{f'g - fg'}{g^2}

f'\big(g(x)\big) \cdot g'(x)

\cos x

-\sin x

\sec^2 x

\sec x \tan x

-\csc x \cot x

-\csc^2 x

e^x

a^x \ln a

\dfrac{1}{x}

\dfrac{1}{x \ln a}

\dfrac{1}{\sqrt{1 - x^2}}

\dfrac{1}{1 + x^2}

12x^3 - 10x

\dfrac{1}{2\sqrt{x}} and -\dfrac{1}{x^2}

6x\cos\left(3x^2\right)

2x\,e^{x^2}

2x \ln x + x

12

y - f(a) = f'(a)(x - a)

A point in the domain where f'(x) = 0 or f'(x) is undefined. Local extrema can only occur there (or at endpoints).

At a critical point: - f' changes + to −: local maximum - f' changes − to +: local minimum - no sign change: neither

f'' > 0 : concave up (holds water) f'' < 0 : concave down An inflection point is where concavity changes.

If f'(c) = 0 : - f''(c) > 0 : local minimum - f''(c) < 0 : local maximum - f''(c) = 0 : test is inconclusive

2x + 2y\dfrac{dy}{dx} = 0 , so \dfrac{dy}{dx} = -\dfrac{x}{y}

v(t) = s'(t) , a(t) = v'(t) = s''(t) Speed is |v(t)| . An object speeds up when v and a have the same sign.

If f is continuous on [a, b] and differentiable on (a, b) , there is a c in (a, b) with f'(c) = \dfrac{f(b) - f(a)}{b - a}

Differentiable implies continuous. The converse is false: f(x) = |x| is continuous at 0 but has a corner there, so it is not differentiable.