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.

Operations

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

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

$(3).\,\,\,$ $\dfrac{d}{dx}{\Big(f(x) \times g(x)\Big)}$ $\,=\,$ $f(x)\dfrac{d}{dx}{g(x)}$ $+$ $g(x)\dfrac{d}{dx}{f(x)}$

$(4).\,\,\,$ $\dfrac{d}{dx}{\Bigg(\dfrac{f(x)}{g(x)}\Bigg)}$ $\,=\,$ $\dfrac{g(x)\dfrac{d}{dx}{f(x)}-f(x)\dfrac{d}{dx}{g(x)}}{g{(x)}^2}$

Reciprocal rule

$\dfrac{d}{dx}{\bigg(\dfrac{1}{f(x)}\bigg)}$ $\,=\,$ $-\,\dfrac{1}{\Big(f(x)\Big)^2} \times \dfrac{d}{dx}{\Big(f(x)\Big)}$

Constant multiple rule

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

Chain rule

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

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.