Calculus Rules

May be cooked

Power Rule

\frac{d}{\differentialD x}\left(x^{n}\right)=nx^{n-1}

Product Rule

\frac{d}{\differentialD x}\left\lbrack f\left(x\right)g\left(x\right)\right\rbrack=f^{\prime}\left(x\right)g\left(x\right)+f\left(x\right)g^{\prime}\left(x\right)

Quotient Rule

\frac{d}{\differentialD x}\left\lbrack\frac{f\left(x\right)}{g\left(x\right)}\right\rbrack=\frac{f^{\prime}\left(x\right)g\left(x\right)-f\left(x\right)g^{\prime}\left(x\right)}{\left(g\left(x\right)\right)^2}

\frac{d}{\differentialD x}\left\lbrack\sin\left(x\right)\right\rbrack

$$=\cos\left(x\right)$$

\frac{d}{\differentialD x}\left\lbrack\cos\left(x\right)\right\rbrack

$$=-\sin\left(x\right)$$

\frac{d}{\differentialD x}\left\lbrack\tan\left(x\right)\right\rbrack

$$=\sec^2\left(x\right)$$

\frac{d}{\differentialD x}\left\lbrack\cot\left(x\right)\right\rbrack

$$=-\csc^2\left(x\right)$$

\frac{d}{\differentialD x}\left\lbrack\sec\left(x\right)\right\rbrack

$$=\sec\left(x_{}\right)\tan\left(x\right)$$

\frac{d}{\differentialD x}\left\lbrack\csc\left(x\right)\right\rbrack

$$=\csc\left(x\right)\cot\left(x\right)$$

$$\lim_{x\to0}\frac{\sin\left(x\right)}{x}$$

$$=1$$

Chain Rule

\frac{d}{\differentialD x}\left\lbrack f\left(g\left(x\right)\right)\right\rbrack=f^{\prime}\left(g\left(x\right)\right)\cdot g^{\prime}\left(x\right)

Average Velocity

$$=\frac{f\left(b\right)-f\left(a\right)}{b-a}$$

Horizontal Asymptope

1. If the degree on top is less than the degree on bottom than y = 0

2. If the degrees are equal, divide the leading coefficients to get asymptope

3. If the degree on top is greater than the degree on bottom there is no horizontal asymptote.

Limit Definition of Derivative

$$\lim_{x\to0}\frac{f\left(x-h\right)-f\left(x\right)}{h}$$