Derivatives with the power rule

Text

Compute the derivatives of the following functions:

• (a) $f(x) = x^{15}$
• (b) $f(x) = 2x^{3}$
• (c) $f(x) = \dfrac{1}{x^{3}}$
• (d) $f(x) = \sqrt{x}$
• (e) $f(x) = \dfrac{1}{\sqrt[3]{x^2}}$
• (f) $f(x) = x^6 - 2x^4 + x^2$

Solution

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

• (a) $f'(x) = 15x^{14}$
• (b) $f'(x) = 6x^{2}$
• (c) $f(x)=x^{-3}\Rightarrow f'(x) = -3x^{-4} = -\dfrac{3}{x^4}$
• (d) $f(x)=x^{1/2}\Rightarrow f'(x) = \dfrac{1}{2}x^{-1/2} = \dfrac{1}{2\sqrt{x}}$
• (e) $f(x)=x^{-2/3}\Rightarrow f'(x) = -\dfrac{2}{3}x^{-5/3} = -\dfrac{2}{3\,x\sqrt[3]{x^2}}$
• (f) $f'(x) = 6x^5 - 8x^3 + 2x$ $\boxed{15x^{14};\ \ 6x^2;\ \ -\tfrac{3}{x^4};\ \ \tfrac{1}{2\sqrt{x}};\ \ -\tfrac{2}{3x\sqrt[3]{x^2}};\ \ 6x^5-8x^3+2x}$