Derivatives of polynomials and constants

Text

Differentiate with respect to $x$ ($a$ and $c$ are constants):

• (a) $y=ax^{3}+7$
• (b) $y=13x^{3/2}-c$
• (c) $y=12x^{1/2}+c^{1/2}$
• (d) $y=c^{1/2}\,x^{1/2}$

Solution

The derivative of a constant is zero; for powers $\left(x^{n}\right)'=n x^{n-1}$.

• (a) $y'=3ax^{2}$
• (b) $y'=\dfrac{39}{2}x^{1/2}$
• (c) $y'=6x^{-1/2}=\dfrac{6}{\sqrt{x}}$
• (d) $y'=\dfrac{1}{2}c^{1/2}x^{-1/2}=\dfrac{\sqrt{c}}{2\sqrt{x}}$ $\boxed{3ax^{2};\ \tfrac{39}{2}\sqrt{x};\ \tfrac{6}{\sqrt{x}};\ \tfrac{\sqrt{c}}{2\sqrt{x}}}$