### 12 - Linearity Rule for differentiation - I

Let $y=kf\left(x\right)$  , where $k$  is a constant. Then

$\frac{dy}{dx}=\underset{h\to 0}{\mathrm{lim}}\frac{kf\left(x+h\right)-kf\left(x\right)}{h}$

$=k\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

$=k{f}^{\prime }\left(x\right)$

Here are few examples where this rule is applied to find the derivatives in such cases.

1. Let $y=3{x}^{4}$  , then $\frac{dy}{dx}=3\frac{d}{dx}\left({x}^{4}\right)=\left(3\right)\left(4{x}^{3}\right)=12{x}^{3}$

2. Let $y=2{x}^{-3}$  , then $\frac{dy}{dx}=2\frac{d}{dx}\left({x}^{-3}\right)=\left(2\right)\left(-3{x}^{-4}\right)=-6{x}^{-4}$

3. $y=4\sqrt{x}=4{x}^{\frac{1}{2}}$  , then $\frac{dy}{dx}=4\frac{d}{dx}\left({x}^{\frac{1}{2}}\right)=\left(4\right)\left(\frac{1}{2}{x}^{-\frac{1}{2}}\right)=2{x}^{\frac{-1}{2}}$