Derivatives of Familiar Functions
Constant Rule
The derivative of a constant is 0.
\[\frac{d}{dx}(c)= 0\]
Linear Function Rule
If \(f(x)=mx+b\), then the derivative is the slope \(m\).
\[\frac{d}{dx}(mx+b)= m\]
Power Rule
\[\frac{d}{dx}(x^n)=n x^{n-1}\]
Trig Rules
\[\frac{d}{dx}\sin(x)=\cos(x)\]
and
\[\frac{d}{dx}\cos(x)=-\sin(x)\]
Exponential Rules
\[\frac{d}{dx}(e^x)=e^x\]
and for any positive number \(a >0\),
\[\frac{d}{dx}(a^x)=(\ln a)a^x\]
Logarithmic Rules
\[\frac{d}{dx}(\ln (x))=\frac{1}{x}\]
and for any positive number \(a >0\),
\[\frac{d}{dx}(\log_a x)= \frac{1}{\ln (a)} \cdot \frac{1}{x}\]
Rules for Combinations of Functions
Product Rule
If \(f(x)\) and \(g(x)\) are functions then
\[(fg)'=f'g+fg' \phantom{hihi} \text{ or } \phantom{hihi}(1st \cdot 2nd )'=D1st \cdot 2nd + 1st \cdot D2nd\]
Quotient Rule
If \(f(x)\) and \(g(x)\) are functions then
\[\left(\frac{f}{g} \right ) '=\frac{gf'-fg'}{g^2} \phantom{hihi}\text{ or } \phantom{hihi}\left(\frac{hi}{lo} \right ) '=\frac{loDhi-hiDlo}{lo^2}\]
Chain Rule
If \(y=f(t)\) and \(t=g(x)\) are functions, then the derivative of \(y=f(g(x))\) is given by
\[\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx} \phantom{hihi}\text{ or } \phantom{hihi}\frac{d}{dx}(f(g(x)))=f'(g(x))g'(x)=D(outside)D(inside)\]