Derivatives with the power rule (2)

Text

Compute the derivative of each function, first rewriting roots and fractions as powers:

• (a) $f(x)=x^{11}$
• (b) $f(x)=x^{-5/2}$
• (c) $f(x)=\sqrt[3]{x^{2}}$
• (d) $f(x)=\dfrac{1}{x^{4}}$
• (e) $f(x)=\sqrt[5]{x^{3}}$
• (f) $f(x)=7x^{2a}$

Solution

Apply the power rule $\left(x^{n}\right)'=n\,x^{n-1}$.

• (a) $f'(x)=11x^{10}$
• (b) $f'(x)=-\dfrac{5}{2}x^{-7/2}$
• (c) $f(x)=x^{2/3}\Rightarrow f'(x)=\dfrac{2}{3}x^{-1/3}=\dfrac{2}{3\sqrt[3]{x}}$
• (d) $f(x)=x^{-4}\Rightarrow f'(x)=-4x^{-5}=-\dfrac{4}{x^{5}}$
• (e) $f(x)=x^{3/5}\Rightarrow f'(x)=\dfrac{3}{5}x^{-2/5}$
• (f) $f'(x)=14a\,x^{2a-1}$ $\boxed{11x^{10};\ -\tfrac{5}{2}x^{-7/2};\ \tfrac{2}{3}x^{-1/3};\ -\tfrac{4}{x^{5}};\ \tfrac{3}{5}x^{-2/5};\ 14a\,x^{2a-1}}$