finding the derivative of ${{\sqrt x}(x^2 - {\sqrt x})}$

Question

I am trying to find the derivative of this expression

${{\sqrt x}(x^2 - {\sqrt x})}$

I would first of simplify the expression to:

${x^{1\over2}(x^2 - x^{1\over2})}$

And then apply ${x^{1\over2}}$ to the term in the brackets"

=> ${x^{3 \over2} - x}$

Then to find the derivative of this expression I get

${{3\over2} x^{1\over2} -1}$

${{3\over2} {\sqrt x} -1}$

But the answer in the book is:

${{5 \over 2}\sqrt[3]{x} - 1}$

The answer in the book is $\frac52x^{\frac32}-1=\frac52x\sqrt{x}-1$ because $x^{\frac12}x^2=x^{\frac52}$, not $x^{\frac32}$. — Did, Jan 04 '16 at 14:18
Your first product is wrong:$x^{\frac{1}{2}}x^2=x^{\frac{5}{2}}$ — sinbadh, Jan 04 '16 at 14:19

Harish Chandra Rajpoot · Accepted Answer · 2016-01-04T15:41:35.750

0

Notice, your mistake $\sqrt x\cdot x^2\ne x^{3/2}$, $$\sqrt x(x^2-\sqrt x)=x^{1/2}(x^2-x^{1/2})=x^{5/2}-x$$ hence, $$\frac{d}{dx}(x^{5/2}-x)=\frac 52x^{\frac 52-1}-1=\color{red}{\frac 52x^{\frac 32}-1}$$

edited Jan 04 '16 at 15:41

answered Jan 04 '16 at 14:22

Harish Chandra Rajpoot

37,450

finding the derivative of ${{\sqrt x}(x^2 - {\sqrt x})}$

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