Constant multiple rule of Derivatives

Formula

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

The derivative of product of a constant and a function is equal to the product of constant and the derivative of the function. This property of differentiation is called the constant multiple rule of derivatives.

Introduction

Let’s take $x$ is a variable, $k$ is a constant and $f(x)$ is a function in terms of $x$. If the constant $k$ is multiplied by the function $f(x)$, then the product of them is $k.f(x)$, which is called as the constant multiple function.

The derivative of the constant multiple function with respect to $x$ is written in mathematical form as follows.

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

The differentiation of the constant multiple function with respect to $x$ is equal to the product of the constant $k$ and the derivative of the function $f(x)$.

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

This property is called the constant multiple rule of differentiation and it is used as a formula in differential calculus.

Examples

Look at the following examples to understand the use of the constant multiple rule in differential calculus.

$(1) \,\,\,$ $\dfrac{d}{dx}{\, \Big(6x^2\Big)} \,=\, 6 \times \dfrac{d}{dx}{\, x^2}$

$(2) \,\,\,$ $\dfrac{d}{dy}{\, \Bigg(\dfrac{\log_{e}{y}}{4}\Bigg)} \,=\, \dfrac{1}{4} \times \dfrac{d}{dy}{\, \log_{e}{y}}$

$(3) \,\,\,$ $\dfrac{d}{dz}{\, \Big(-0.7\sin{3z}\Big)} \,=\, -0.7 \times \dfrac{d}{dx}{\, \sin{3z}}$

Proof

Learn how to derive the constant multiple rule in differential calculus.