Derivative of a chain of three functions

Text

Differentiate the function $y=\sqrt{(3x-1)^{4}+5}$.

Solution

Let $u=(3x-1)^{4}+5$, so $y=u^{1/2}$. $u'=4(3x-1)^{3}\cdot3=12(3x-1)^{3}$. $y'=\dfrac{u'}{2\sqrt{u}}=\dfrac{6(3x-1)^{3}}{\sqrt{(3x-1)^{4}+5}}$.