# 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.

Latest Math Topics
Email subscription
Math Doubts is a best place to learn mathematics and from basics to advanced scientific level for students, teachers and researchers. Know more
###### Math Problems

Learn how to solve easy to difficult mathematics problems of all topics in various methods with step by step process and also maths questions for practising.