User:Barrucadu/ListOfDerivatives

General differentiation rules

 * Linearity
 * $$\left({cf}\right)' = cf'$$
 * $$\left({f + g}\right)' = f' + g'$$


 * Product rule
 * $$\left({fg}\right)' = f'g + fg'$$


 * Reciprocal rule
 * $$\left(\frac{1}{f}\right)' = \frac{-f'}{f^2}, \qquad f \ne 0$$


 * Quotient rule
 * $$\left({f \over g}\right)' = {f'g - fg' \over g^2}, \qquad g \ne 0$$


 * Chain rule
 * $$(f \circ g)' = (f' \circ g)g'$$


 * Derivative of inverse function
 * $$(f^{-1})' =\frac{1}{f' \circ f^{-1}}$$

for any differentiable function f of a real argument and with real values, when the indicated compositions and inverses exist.
 * Generalized power rule
 * $$(f^g)'=f^g \left( g'\ln f + \frac{g}{f} f' \right)$$


 * Derivative of implicit function
 * If implicit function $$y(x)$$ is defined as $$F(x,y(x))=0$$
 * then
 * $$y'_x=-\frac{F'_x}{F'_y}=-\frac{\partial F}{\partial x}/\frac{\partial F}{\partial y}$$


 * Derivative of parametrically defined function
 * If a function $$y(x)$$ defined parametrically

\begin{cases} x = f(t)\\ y = g(t) \end{cases} $$
 * then
 * $$y'_x=\frac{g'(f^{-1}(x))}{f'(f^{-1}(x))}$$


 * Derivative of complex function.
 * For complex function $$z(t)=F(u(t),v(t))$$
 * the full derivative is
 * $$z'=F'_u u'+F'_v v'=\frac{\partial F}{\partial u}u'+\frac{\partial F}{\partial v} v'\,$$
 * If $$z(x)=F(x,y(x))$$
 * Then full derivative is
 * $$z'=F'_x +F'_y y'=\frac{\partial F}{\partial x}+\frac{\partial F}{\partial y} y'\,$$

Derivatives of simple functions

 * $$c' = 0 \, $$


 * $$x' = 1 \, $$


 * $$(cx)' = c \, $$


 * $$|x|' = {x \over |x|} = \sgn x,\qquad x \ne 0$$


 * $$(x^c)' = cx^{c-1} \qquad \mbox{where both } x^c \mbox{ and } cx^{c-1} \mbox { are defined}$$


 * $$\left({1 \over x}\right)' = \left(x^{-1}\right)' = -x^{-2} = -{1 \over x^2}$$


 * $$\left({1 \over x^c}\right)' = \left(x^{-c}\right)' = -cx^{-(c+1)} = -{c \over x^{c+1}}$$


 * $$\left(\sqrt{x}\right)' = \left(x^{1\over 2}\right)' = {1 \over 2} x^{-{1\over 2}}  = {1 \over 2 \sqrt{x}}        , \qquad x > 0$$

Derivatives of exponential and logarithmic functions

 * $$ \left(c^x\right)' = {c^x \ln c } ,\qquad c > 0$$

note that the equation above is true for all c, but the derivative yields a complex number.


 * $$ \left(e^x\right)' = e^x$$


 * $$ \left( \log_c x\right)' = {1 \over x \ln c}, \qquad c > 0, c \ne 1$$

the equation above is also true for all c but yields a complex number.


 * $$ \left( \ln x\right)' = {1 \over x}, \qquad x \ne 0$$


 * $$ \left( \ln |x|\right)' = {1 \over x}$$


 * $$ \left( x^x \right)' = x^x(1+\ln x)$$

The derivative of the natural logarithm with a generalised functional argument f(x) is


 * $$ \frac{d}{dx}[\ln(f(x))] = \frac{f'(x)}{f(x)}$$

By applying the change-of-base rule, the derivative for other bases is


 * $$\frac{d}{dx} \log_b(x) = \frac{d}{dx} \frac {\ln(x)}{\ln(b)} = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}.$$