Derivative properties

There are six properties in differential calculus and they are used as formulas in differentiation. So, learn the following list of properties of derivatives with proofs and also example problems with solutions to learn how to use them in differentiating the functions.

Sum rule

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

Difference rule

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

Product rule

The product rule of derivatives is written popularly in two different forms.

$(1). \,\,\,$ $\dfrac{d}{dx}{\, \Big(f{(x)}.g{(x)}\Big)}$ $\,=\,$ ${f{(x)}}{\dfrac{d}{dx}{\, g{(x)}}}$ $+$ ${g{(x)}}{\dfrac{d}{dx}{\, f{(x)}}}$

$(2). \,\,\,$ $\dfrac{d}{dx}{\, \Big(u.v\Big)}$ $\,=\,$ $u.\dfrac{dv}{dx}+v.\dfrac{du}{dx}$

Quotient rule

The quotient rule of differentiation is also written popularly in two different forms.

$(1). \,\,\,$ $\dfrac{d}{dx}{\, \Bigg(\dfrac{f{(x)}}{g{(x)}}\Bigg)}$ $\,=\,$ $\dfrac{{g{(x)}}{\dfrac{d}{dx}{f{(x)}}}-{f{(x)}}{\dfrac{d}{dx}{g{(x)}}}}{{g{(x)}}^2}$

$(2). \,\,\,$ $\dfrac{d}{dx}{\, \Bigg(\dfrac{u}{v}\Bigg)}$ $\,=\,$ $\dfrac{v.\dfrac{du}{dx}-u.\dfrac{dv}{dx}}{v^2}$

Constant multiple rule

$\dfrac{d}{dx}{\, \Big(k.f(x)\Big)} \,=\, k \times \dfrac{d}{dx}{\, f(x)}$

Chain rule

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

Formulas

List of the differentiation formulas with proofs and example problems to learn how to use some standard results as formulas in differentiating the functions.