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Constant Multiple Rule

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The Constant Multiple Rule. If c{c} is a constant anf f{f{}} is a differentiable function then (cf(x))′=c(f(x))′{\left({c}{f{{\left({x}\right)}}}\right)}'={c}{\left({f{{\left({x}\right)}}}\right)}'.

Proof. By definition (cf(x))′=lim⁡h→0cf(x+h)−cf(x)h=clim⁡h→0f(x+h)−f(x)h=cf′(x){\left({c}{f{{\left({x}\right)}}}\right)}'=\lim_{{{h}\to{0}}}\frac{{{c}{f{{\left({x}+{h}\right)}}}-{c}{f{{\left({x}\right)}}}}}{{h}}={c}\lim_{{{h}\to{0}}}\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}={c}{f{'}}{\left({x}\right)}.

Example 1. Find f′(x){f{'}}{\left({x}\right)} if f(x)=2⋅3x{f{{\left({x}\right)}}}={2}\cdot{{3}}^{{x}}.

f′(x)=(2⋅3x)′=2(3x)′=2⋅3xln⁡(3){f{'}}{\left({x}\right)}={\left({2}\cdot{{3}}^{{x}}\right)}'={2}{\left({{3}}^{{x}}\right)}'={2}\cdot{{3}}^{{x}}{\ln{{\left({3}\right)}}}.

Example 2. Find f′(x){f{'}}{\left({x}\right)} if f(x)=12x{f{{\left({x}\right)}}}=\frac{{1}}{{{2}\sqrt{{{x}}}}}.

f′(x)=(12x)′=12(1x)′=12(1x12)′=12(x−12)′=12⋅(−12)x−12−1=−14x−32{f{'}}{\left({x}\right)}={\left(\frac{{1}}{{{2}\sqrt{{{x}}}}}\right)}'=\frac{{1}}{{2}}{\left(\frac{{1}}{\sqrt{{{x}}}}\right)}'=\frac{{1}}{{2}}{\left(\frac{{1}}{{{{x}}^{{\frac{{1}}{{2}}}}}}\right)}'=\frac{{1}}{{2}}{\left({{x}}^{{-\frac{{1}}{{2}}}}\right)}'=\frac{{1}}{{2}}\cdot{\left(-\frac{{1}}{{2}}\right)}{{x}}^{{-\frac{{1}}{{2}}-{1}}}=-\frac{{1}}{{4}}{{x}}^{{-\frac{{3}}{{2}}}}.