Operations of the Derivatives

The functions, which involve in basic mathematical operations are appeared in differential calculus. The derivatives of such functions require some special properties to find the differentiation. The following four properties are useful to find the derivatives of the functions, which involve in fundamental operations. So, let’s learn the following differentiation properties with proofs and example problems.

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

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

Quotient Rule

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