# Constant Multiple Rule

## Related calculator: Online Derivative Calculator with Steps

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

Proof. By definition ${\left({c}{f{{\left({x}\right)}}}\right)}'=\lim_{{{h}\to{0}}}\frac{{{c}{f{{\left({x}+{h}\right)}}}-{c}{f{{\left({x}\right)}}}}}{{h}}={c}\lim_{{{h}\to{0}}}\frac{{{f{{\left({x}+{h}\right)}}}-{f{{\left({x}\right)}}}}}{{h}}={c}{f{'}}{\left({x}\right)}$.

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

${f{'}}{\left({x}\right)}={\left({2}\cdot{{3}}^{{x}}\right)}'={2}{\left({{3}}^{{x}}\right)}'={2}\cdot{{3}}^{{x}}{\ln{{\left({3}\right)}}}$.

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

${f{'}}{\left({x}\right)}={\left(\frac{{1}}{{{2}\sqrt{{{x}}}}}\right)}'=\frac{{1}}{{2}}{\left(\frac{{1}}{\sqrt{{{x}}}}\right)}'=\frac{{1}}{{2}}{\left(\frac{{1}}{{{{x}}^{{\frac{{1}}{{2}}}}}}\right)}'=\frac{{1}}{{2}}{\left({{x}}^{{-\frac{{1}}{{2}}}}\right)}'=\frac{{1}}{{2}}\cdot{\left(-\frac{{1}}{{2}}\right)}{{x}}^{{-\frac{{1}}{{2}}-{1}}}=-\frac{{1}}{{4}}{{x}}^{{-\frac{{3}}{{2}}}}$.