Use the rules for derivatives to find the derivatives below. y=\sqrt{x}-x

Anonym 2021-10-01 Answered
Use the rules for derivatives to find the derivatives below.
\(\displaystyle{y}=\sqrt{{{x}}}-{x}^{{{2}}}\); use the Power Rule

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Expert Answer

Ayesha Gomez
Answered 2021-10-02 Author has 10698 answers
\(\displaystyle{f{{\left({x}\right)}}}={y}_{{{1}}}=\sqrt{{{x}}}-{x}^{{{2}}}\)
we have to find the derivative by using the power rule:
Power Rule:
\(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({x}\right)}^{{{n}}}={n}{\left({x}\right)}^{{{n}-{1}}}\)
Now
\(\displaystyle{y}_{{{1}}}={x}^{{{\frac{{{1}}}{{{2}}}}}}-{x}^{{{2}}}\)
\(\displaystyle{f}'{\left({x}\right)}={y}_{{{1}}}'={\frac{{{1}}}{{{2}}}}{x}^{{{\frac{{{1}}}{{{2}}}}-{1}}}-{2}{x}^{{{2}-{1}}}\)
\(\displaystyle={\frac{{{1}}}{{{2}}}}{x}^{{-{\frac{{{1}}}{{{2}}}}}}-{2}{x}\)
\(\displaystyle={\frac{{{1}}}{{{2}\sqrt{{{x}}}}}}-{2}{x}\)
Hence, \(\displaystyle{f}'{\left({x}\right)}={y}_{{{1}}}'={\frac{{{1}}}{{{2}\sqrt{{{x}}}}}}-{2}{x}\)
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