Differentiation#
Derivatives#
Description |
Equations |
---|---|
Derivative as a function |
\(f'(x) = \lim\limits_{h \to 0}\dfrac{f(x+h) - f(x)}{h}\) |
Geometric interpretation of derivatives |
The tangent line of \(y = f(x)\) at \((a, f(a))\) has a slope of \(f'(a)\) |
Derivatives and instantaneous rate of change |
The derivative \(f'(a)\) is the instantaneous rate of change of \(y=f(x)\) with respect to \(x\) when \(x=a\): |
Differentiation and continuity |
If \(f\) is differentiable at \(a\), |
Non-differentiable conditions |
1. a corner |
Differentiation rules#
Description |
Equations |
---|---|
Constant multiple rule |
\(\frac{d}{dx}[cf(x)] = c\frac{d}{dx}f(x)\) |
Addition and subtraction rule |
\(\frac{d}{dx}[f(x)\pm g(x)] = \frac{d}{dx}f(x) \pm \frac{d}{dx}g(x)\) |
Product rule |
\(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\) |
Quotient rule |
\(\frac{d}{dx}\dfrac{f(x)}{g(x)} = \dfrac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}\) |
Constant rule |
\(\frac{d}{dx}c = 0\) |
Power rule |
\(\frac{d}{dx} x^n = nx^{n-1}\) |
Chain rule |
\(\dfrac{dy}{dx} = \dfrac{dy}{du}\dfrac{du}{dx} \newline \frac{d}{dx} f(g(x)) = f'(g(x))\cdot g'(x)\) |
Linear approximations |
\(f(x) \approx f(a) + f'(a)(x-a)\) |
Differentials |
\(dy = f'(x) \ dx\) |
Special limits#
Description |
Equations |
---|---|
Limit associated with sine |
\(\lim\limits_{\theta \to 0}\dfrac{\sin\theta}{\theta} = 1\) |
Limit associated with cosine |
\(\lim\limits_{\theta\to 0}\dfrac{\cos\theta - 1}{\theta} = 0\) |
Definition of \(e\) |
\(\lim\limits_{h\to 0}\dfrac{e^h - 1}{h} = 1\) |
\(e\) as a limit |
\(e = \lim\limits_{x\to 0} (1+x)^{1/x}\) |
\(e\) as a limit |
\(e = \lim\limits_{n\to \infin} (1+\frac{1}{n})^{n}\) |
Table of derivatives#
Function \(f(x)\) |
Derivative \(f'(x)\) |
Function \(f(x)\) |
Derivative \(f'(x)\) |
---|---|---|---|
\(c\) |
\(0\) |
\(x^n\) |
\(nx^{n-1}\) |
\(e^x\) |
\(e^x\) |
\(\ln x\) |
\(\dfrac{1}{x}\) |
\(\sin x\) |
\(\cos x\) |
\(\cos x\) |
\(-\sin x\) |
\(\tan x\) |
\(\sec^2 x\) |
\(\arcsin x\) |
\(\dfrac{1}{\sqrt{1-x^2}}\) |
\(\arctan x\) |
\(\dfrac{1}{1+x^2}\) |