AP Calculus AB Derivative Rules

All the derivative rules you could possibly need

$$s(t)$$

$$s\left(t\right)=\frac12gt^2+v_0t+s_0$$

$$v(t)$$

$v(t) = s'(t)$ $\frac{lim}{\Delta x\to0}\frac{(s(\Delta t+t)-s(t)}{\Delta t}$

V ave

$$v_{avg}=\frac{\Delta s}{\Delta t}$$

m

$$m=f^{\prime}(a)=\lim_{\Delta x\to0}\frac{f(c+\Delta x)-f(c)}{\Delta x}=\lim_{\Delta x\to0}\frac{\Delta y}{\Delta x}$$

$$\frac{d}{dx}[c]$$

$\frac{d}{dx}[c] = 0$ .

$$\frac{d}{dx}[x^{n}]$$

$\frac{d}{dx}[x^n] = n x^{n-1}$ .

$$\frac{d}{dx}[c\cdot f(x)]$$

$\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)$ .

$$\frac{d}{dx}[cx^{n}]$$

$\frac{d}{dx}[c x^n] = c \cdot n x^{n-1}$ .

$$\frac{d}{dx}[f(x)+g(x)]$$

$\frac{d}{dx}[f(x)+g(x)]=f^{\prime}(x)+g^{\prime}(x)$ .

$$\frac{d}{dx}[e^{x}]$$

$$\frac{d}{dx}[e^x] = e^x$$

$$\frac{d}{dx}[\sin(x)]$$

$$\frac{d}{dx}[\sin(x)] = \cos(x)$$

$$\frac{d}{dx}[\cos(x)]$$

$$\frac{d}{dx}[\cos(x)] = -\sin(x)$$

$$f\text{'}(x)$$

$$f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}$$

$$\frac{d}{dx}[f(x)\cdot g(x)]$$

$$\frac{d}{dx}[f(x)\cdot g(x)]=f^{}(x)g^{\prime}(x)+g^{}(x)f^{\prime}\left(x\right)$$

Quotient Rule

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f^{\prime}(x)-f(x)g^{\prime}(x)}{[g(x)]^2}$$

$$\frac{d}{dx}[\tan(x)]$$

$$\frac{d}{dx}[\tan(x)] = \sec^2(x)$$

$$\frac{d}{dx}[\csc(x)]$$

$$\frac{d}{dx}[\csc(x)] = -\csc(x)\cot(x)$$

$$\frac{d}{dx}[\sec(x)]$$

$$\frac{d}{dx}[\sec(x)] = \sec(x)\tan(x)$$

$$\frac{d}{dx}[\cot(x)]$$

$$d/dx[cot(x)]=-csc^2x$$

$$\frac{d}{dx}[f(x)-g(x)]$$

$$\frac{d}{dx}[f(x)-g(x)]=f^{\prime}(x)-g^{\prime}\left(x\right)$$

$$\frac{d}{dx}[f(g(x))]$$

$$dy/dx\left\lbrack{f(g(x))]=f^{\prime}(g(x))g^{\prime}(x)}\right.$$