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

Use the rules for derivatives to find the derivatives below.
$$\displaystyle{y}=\sqrt{{{x}}}-{x}^{{{2}}}$$; use the Power Rule

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Ayesha Gomez
$$\displaystyle{f{{\left({x}\right)}}}={y}_{{{1}}}=\sqrt{{{x}}}-{x}^{{{2}}}$$
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, $$\displaystyle{f}'{\left({x}\right)}={y}_{{{1}}}'={\frac{{{1}}}{{{2}\sqrt{{{x}}}}}}-{2}{x}$$