AP Calculus AB Derivative Rules

All the derivative rules you could possibly need

Position function $s(t)$

$$\frac12gt^2+v_0t+s_0$$

Velocity function $v(t)$

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

Average velocity

$$\frac{\Delta s}{\Delta t}$$

Tangent line to $f(x)$ at $(a, f(a))$ with slope $m$

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

The Constant Rule

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

The Power Rule

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

The Constant Multiple Rule

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

Combined Constant Multiple and Power Rule

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

Sum Rule

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

Derivative of the natural exponential function

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

Derivatives of sine function

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

Derivation of cosine function

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

Definition of the derivative of a function $f(x)$

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

Product Rule

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

Quotient Rule

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

Derivative of tangent function

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

Derivative of cosecant function

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

Derivative of secant function

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

Derivative of cotangent function

\frac{d}{\differentialD x}[cot(x)]=-csc^2x

Difference Rule

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