3. Differentiation Rules

Standard

$(x^r{)}^{\prime }=r{x}^{r-1}$ for any real value $r$ (power of $x$)

$(e^x{)}^{\prime }=e^x$ (exponential function)

$[e^{u(x)}{]}^{\prime }=e^{u(x)}\cdot u^{\prime }(x)$ (chain rule + exponential function)

$(\ln x{)}^{\prime }=\frac{1}{x}$ (natural logarithm)

$[\ln (u(x)){]}^{\prime }=\frac{1}{u(x)}\cdot u^{\prime }(x)$ (chain rule + natural logarithm)

$(\sin (x){)}^{\prime }=\cos (x)$ (sine function)

$[\sin (u(x)){]}^{\prime }=\cos (u(x))\cdot u^{\prime }(x)$ (chain rule + sine function)

$(\cos (x){)}^{\prime }=-\sin (x)$ (cosine function)

$[\cos (u(x)){]}^{\prime }=-\sin (u(x))\cdot u^{\prime }(x)$ (chain rule + cosine function)

Computation

$(cf(x){)}^{\prime }=c{f}^{\prime }(x)$ for any real value $c$

$(f(x)+g(x){)}^{\prime }=f^{\prime }(x)+g^{\prime }(x)$ (Sum Rule)

$(f(x)g(x){)}^{\prime }=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)$ (Product Rule)

$(\frac{f(x)}{g(x)}{)}^{\prime }=\frac{f^{\prime }(x)g(x)-f(x)g^{\prime }(x)}{(g(x){)}^2}$ (Quotient Rule)

$f(g(x){)}^{\prime }=f^{\prime }(g(x))g^{\prime }(x)$ (Chain Rule)