constant multiple rule | derivatives,ab calculus,12th grade

# Constant Multiple Rule

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The constant multiple rule : If 'f' is a differentiable function and k is any real number, then kf is also differentiable and

$\frac{\text{d}[kf(x)]}{\text{d}x}=k\frac{\text{d}f(x)}{\text{d}x}= kf'(x)$

Prove that: $\frac{\text{d}[kf(x)]}{\text{d}x}= kf'(x)$

Proof : $\frac{\text{d}[kf(x)]}{\text{d}x} = \lim_{\triangle x \rightarrow 0}\frac{k f(x+\triangle x)-kf(x)}{\triangle x}$

= $\lim_{\triangle x \rightarrow 0}k[\frac{ f(x+\triangle x)-f(x)}{\triangle x}]$

$\frac{\text{d}[kf(x)]}{\text{d}x}=k[\lim_{\triangle x \rightarrow 0}\frac{ f(x+\triangle x)-f(x)}{\triangle x}]$

According to the definition of derivative
$\frac{\text{d}[f(x)]}{\text{d}x} = \lim_{\triangle x \rightarrow 0}\frac{f(x+\triangle x)-f(x)}{\triangle x}$

So, $\frac{\text{d}[kf(x)]}{\text{d}x} = k\frac{\text{d}[f(x)]}{\text{d}x}$

1) $\frac{\text{d}[kf(x)]}{\text{d}x} = k\frac{\text{d}[f(x)]}{\text{d}x}$

2) ${\frac{\text{d}[\frac{f(x)}{k}]}{\text{d}x} = \frac{\text{d}[\frac{1}{k}f(x)]}{\text{d}x}}=\frac{1}{k}\frac{\text{d}[f(x)]}{\text{d}x}$

## Examples of constant multiple rule

1) Find : $\frac{\text{d}[5x^{4}]}{\text{d}x}$

Solution : $\frac{\text{d}[5x^{4}]}{\text{d}x}$

=$5\frac{\text{d}x^{4}}{\text{d}x}$

=$5x^{4-1}$
$\frac{\text{d}[5x^{4}]}{\text{d}x}=5x^{3}$

2)
Find : $\frac{\text{d}[\frac{3}{x}]}{\text{d}x}$

Solution : $\frac{\text{d}[\frac{3}{x}]}{\text{d}x}$

= 3$\frac{\text{d}[\frac{1}{x}]}{\text{d}x}$

=3$\frac{\text{d}[x^{-1}]}{\text{d}x}$

=3$x^{-1-1}$

=3$x^{-2}$

=3$\frac{1}{x^{2}}$

$\frac{\text{d}[\frac{3}{x}]}{\text{d}x} =\frac{3}{x^{2}}$

3)
Find: $\frac{\text{d}[\frac{5}{\sqrt[3]{x^{2}}}]}{\text{d}x}$

=$\frac{\text{d}[\frac{5}{\sqrt[3]{x^{2}}}]}{\text{d}x}$

=5$\frac{\text{d}[\frac{1}{\sqrt[3]{x^{2}}}]}{\text{d}x}$

=5$\frac{\text{d}[\frac{1}{{x^{\frac{2}{3}}}}]}{\text{d}x}$

=5$\frac{\text{d}[{{x^{\frac{-2}{3}}}}]}{\text{d}x}$

=-5$x^{(\frac{-2}{3}-1)}$

=-5$x^{\frac{-5}{3}}$

= -5$\frac{1}{x^{\frac{5}{3}}}$

$\frac{\text{d}[\frac{5}{\sqrt[3]{x^{2}}}]}{\text{d}x}=-\frac{5}{x^{\frac{5}{3}}}$

The constant multiple rule and the power Rule can be combined into one rule. The combination rule is given by

$\color{red}{\frac{\text{d}(kx^{n})}{\text{d}x}=knx^{n-1}}$

Practice questions :
1) Find the derivative of $\frac{-3x^{2}}{2}$

2) Find the derivative of $\frac{7}{2x^{3}}$

3) Find the derivative of $16x^{5}$

4) Find the derivative of $6x^{-3}$

5) Find the derivative of $\frac{1}{2x^{-2}}$