Derivatives with the power rule (3)

Text

Compute the derivatives of the following functions:

• (a) $f(x)=x^{9}$
• (b) $f(x)=3x^{4}$
• (c) $f(x)=\dfrac{1}{x^{5}}$
• (d) $f(x)=\sqrt[3]{x}$
• (e) $f(x)=x^{7/2}$
• (f) $f(x)=2x^{5}-3x^{3}+x$

Solution

Apply the power rule $\left(x^n\right)'=n\,x^{n-1}$, first rewriting roots and fractions as powers.

• (a) $f'(x)=9x^{8}$
• (b) $f'(x)=12x^{3}$
• (c) $f(x)=x^{-5}\Rightarrow f'(x)=-5x^{-6}=-\dfrac{5}{x^{6}}$
• (d) $f(x)=x^{1/3}\Rightarrow f'(x)=\dfrac{1}{3}x^{-2/3}=\dfrac{1}{3\sqrt[3]{x^{2}}}$
• (e) $f'(x)=\dfrac{7}{2}x^{5/2}$
• (f) $f'(x)=10x^{4}-9x^{2}+1$ $\boxed{9x^{8};\ \ 12x^{3};\ \ -\tfrac{5}{x^{6}};\ \ \tfrac{1}{3\sqrt[3]{x^{2}}};\ \ \tfrac{7}{2}x^{5/2};\ \ 10x^{4}-9x^{2}+1}$