Differentiation rules and standard derivatives

    \begin{align*} &\text{function} && \text{derivative}\\ & x^r && rx^{r-1}\\ & e^x && e^x\\ & a^x && \ln(a)a^x\\ & \ln(x) && \frac{1}{x},\qquad x > 0\\ & \log_a(x) && \frac{1}{x\ln(a)}\\ & \sin(x) && \cos(x)\\ & \cos(x) && -\sin(x)\\ &\tan(x) && \sec^2(x) = \frac{1}{\cos^2(x)} = 1+\tan^2(x)\\ &\arcsin(x) && \frac{1}{\sqrt{1-x^2}}, -1<x<1\\ &\arccos(x) && -\frac{1}{\sqrt{1-x^2}}, -1<x<1\\ &\arctan(x) && \frac{1}{1+x^2}\\ \end{align*}

Differentiation rules
NB f and g are functions of x, f(x) and g(x).

    \begin{align*} &\text{name} && \text{function} && \text{derivative}\\ &\text{product rule} &&fg && f'g+fg'\\ &\text{chain rule} &&f(g(x)) &&f'(g(x))g'(x) \\ &\text{reciprocal rule }&& \frac{1}{f(x)}&& -\frac{f'(x)}{(f(x))^2} \\ &\text{quotient rule} &&f(x)/g(x) &&\frac{f'g-g'f}{g^2}, g\neq 0 \\ &\text{generalized power rule} &&f^g =e^{g\ln{f}} &&f^g(f'\frac{g}{f}+g'\ln{f}) \end{align*}

Leave a Reply

Your email address will not be published.