Derivatives

The following is a table of first order derivatives.

Description	Derivative	Result
Constant	$c^{\prime }$	$0$
Variable	$x^{\prime }$	$1$
Scaled variable	$(cx{)}^{\prime }$	$c{x}^{\prime }$
Sum	$(x+y{)}^{\prime }$	$x^{\prime }+y^{\prime }$
Product	$(xy{)}^{\prime }$	$x{y}^{\prime }+x^{\prime }y$
Quot	$\frac{x}{y}$	$\frac{x.y^{\prime }+x^{\prime }.y}{y^2}$
Reciprocal	$\frac{1}{x}$	$\frac{-x^{\prime }}{x^2}$
Power	$(x^y{)}^{\prime }$	$y{x}^{y-1}$
Square root	${\sqrt{x}}^{\prime }$	$\frac{1}{2\sqrt{x}}$
Chained rule	$\frac{\partial f(g(x))}{\partial x}$	$\frac{\partial f(g(x))}{\partial g(x)} \frac{\partial g(x)}{\partial x} $
Multivariable rule	$\frac{\partial f(u(x),v(x))}{\partial x}$	$\frac{\partial f(u(x),v(x))}{\partial u(x)}\frac{\partial f(u(x),v(x))}{\partial u(x)}+\frac{\partial f(u(x),v(x))}{\partial x}$
Vector square length	$(\overrightarrow{v}.\overrightarrow{v}{)}^{\prime }$	$2\overrightarrow{v}{\overrightarrow{v}}^{\prime }$
Vector length	$\Vert \overrightarrow{v}{\Vert }^{\prime }$	$\hat{v}{\overrightarrow{v}}^{\prime }$