Derivatives

The following is a table of first order derivatives.

Description	Derivative	Result
Constant	$c'$	$0$
Variable	$x'$	$1$
Scaled variable	$(cx)'$	$cx'$
Sum	$(x+y)'$	$x' + y'$
Product	$(xy)'$	$xy'+x'y$
Quot	$\frac{x}{y}$	$\frac{x.y'+x'.y}{y^2}$
Reciprocal	$\frac{1}{x}$	$\frac{-x'}{x^2}$
Power	$(x^y)'$	$yx^{y-1}$
Square root	$\sqrt{x}'$	$\frac{1}{2\sqrt{x}}$
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	$(\vec{v}.\vec{v})'$	$2\vec{v}\vec{v}'$
Vector length	$\\|\vec{v}\\|'$	$\hat{v}\vec{v}'$

Description	Derivative	Result
Constant	\(c'\)	\(0\)
Variable	\(x'\)	\(1\)
Scaled variable	\((cx)'\)	\(cx'\)
Sum	\((x+y)'\)	\(x' + y'\)
Product	\((xy)'\)	\(xy'+x'y\)
Quot	\(\frac{x}{y}\)	\(\frac{x.y'+x'.y}{y^2}\)
Reciprocal	\(\frac{1}{x}\)	\(\frac{-x'}{x^2}\)
Power	\((x^y)'\)	\(yx^{y-1}\)
Square root	\(\sqrt{x}'\)	\(\frac{1}{2\sqrt{x}}\)
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	\((\vec{v}.\vec{v})'\)	\(2\vec{v}\vec{v}'\)
Vector length	\(\\|\vec{v}\\|'\)	\(\hat{v}\vec{v}'\)