Unit 2 Derivatives

Rules and Definition

f^{\prime}\left(x\right)

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

f^{\prime}\left(c\right)

\lim_{x\to c}\frac{f\left(x\right)-f\left(c\right)}{x-c}

Average Rate of Change

The slope over an interval

Instantaneous Rate of Change

The slope at a point (derivative)

Product Rule

f\cdot g^{\prime}+g\cdot f^{\prime}

Quotient Rule

\left(\frac{f}{g}\right)=\frac{g\cdot f^{\prime}-f\cdot g^{\prime}}{g^2}

\left(x^{n}\right)

nx^{\left(n-1\right)}

\left(e^{x}\right)

e^{x}

\left(\ln x\right)

\frac{1}{x}

\left(\sin x_{}\right)

\cos x

\left(\cos x\right)

-\sin x

\left(\tan x\right)

\sec^2x

\left(\csc x\right)

-\csc x\cot x

\left(\sec x\right)

\sec x\tan x

\left(\cot x\right)

-\csc^2x

x^{-n}

\frac{1}{x^{n}}

x^{\frac{m}{n}}

\sqrt[n]{x^{m}}

Equation

y-\left(f\left(x\right)\right)=x\left(derivative\right)\cdot\left(x-initial\right)