Problem 2-11: Differentiating by Chain Rule


Differentiate  \left(1+\sqrt{x}\right)^{8} by using the chain rule.


Solution
Chain rule:  \displaystyle\frac{du}{dx}=\displaystyle\frac{du}{dv}\displaystyle\frac{dv}{dx}

We must determine the individual components of the chain rule to apply it. Set  u= \left(1+\sqrt{x}\right)^{8} and  v=1+\sqrt{x}.

Therefore,
 u=v^8,
 \displaystyle\frac{du}{dv}=8v^7,

 \displaystyle\frac{dv}{dx}=\displaystyle\frac{d}{dx}x^{\frac{1}{2}}=\displaystyle\frac{1}{2}x^{\frac{1}{2}-1}=\displaystyle\frac{1}{2}x^{-\frac{1}{2}}=\displaystyle\frac{1}{2\sqrt{x}}.

Hence,
 \displaystyle\frac{du}{dx}=\displaystyle\frac{du}{dv}\displaystyle\frac{dv}{dx}=\left(8v^7\right)\left(\displaystyle\frac{1}{2\sqrt{x}}\right)=\displaystyle\frac{4\left(1+\sqrt{x}\right)^7}{\sqrt{x}}

Problem 2-11

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