Question

find the derivative of the function. y = \sqrt9{x^{2}}-x^{\sqrt{2}}

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt[9]{x^{2}}-x^{\sqrt{2}}$ as $y=x^{\frac{2}{9}}-x^{\sqrt{2}}$.

Step2: Apply power - rule for derivatives

The power - rule states that if $y = x^{n}$, then $y'=nx^{n - 1}$. For the first term $x^{\frac{2}{9}}$, its derivative is $\frac{2}{9}x^{\frac{2}{9}-1}=\frac{2}{9}x^{-\frac{7}{9}}=\frac{2}{9x^{\frac{7}{9}}}$. For the second term $x^{\sqrt{2}}$, its derivative is $\sqrt{2}x^{\sqrt{2}-1}$.

Step3: Write the derivative of the function

$y'=\frac{2}{9x^{\frac{7}{9}}}-\sqrt{2}x^{\sqrt{2}-1}$.

Answer:

$y'=\frac{2}{9x^{\frac{7}{9}}}-\sqrt{2}x^{\sqrt{2}-1}$