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Derivative properties

Sum rule

$\dfrac{d}{dx}{\, \Big(f{(x)}+g{(x)}\Big)}$ $\,=\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $+$ $\dfrac{d}{dx}{\, g{(x)}}$

The derivative of sum of two or more functions is equal to sum of their derivatives.

Difference rule

$\dfrac{d}{dx}{\, \Big(f{(x)}-g{(x)}\Big)}$ $\,=\,$ $\dfrac{d}{dx}{\, f{(x)}}$ $-$ $\dfrac{d}{dx}{\, g{(x)}}$

The derivative of difference of functions is equal to difference of their derivatives.

Product rule

$\dfrac{d}{dx}{\, \Big(f{(x)}.g{(x)}\Big)}$ $\,=\,$ ${f{(x)}}{\dfrac{d}{dx}{\, g{(x)}}}$ $+$ ${g{(x)}}{\dfrac{d}{dx}{\, f{(x)}}}$

The derivative of product of two functions is equal to sum of product of first function and derivative of second function, and product of second function and derivative of first function.

Quotient rule

$\dfrac{d}{dx}{\, \Bigg(\dfrac{f{(x)}}{g{(x)}}\Bigg)}$ $\,=\,$ $\dfrac{{g{(x)}}{\dfrac{d}{dx}{f{(x)}}}-{f{(x)}}{\dfrac{d}{dx}{g{(x)}}}}{{g{(x)}}^2}$

The derivative of quotient of functions is equal to quotient of subtraction of product of function in numerator and derivative of function in denominator from the product of function in denominator and derivative of function in numerator by the square of function in denominator.

Chain rule

$\dfrac{d}{dx} {f[{g(x)}]} \,=\, {f'[{g(x)}]}.{g'{(x)}}$