Constant Multiple Rule
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 $\frac{1}{2x^{-2}}$
12th grade math
Home