find the derivative (differentiate)\n1. $y = sqrt{x^{3}-x^{2}+5x}$\n$y=(x^{3}-x^{2}+5x)^{\frac{1}{2}}$\n$y=\f…

Explanation:

Step1: Rewrite the function

Rewrite $y = \sqrt{x^{3}-x^{2}+5x}$ as $y=(x^{3}-x^{2}+5x)^{\frac{1}{2}}$ using the power - root relationship $\sqrt{u}=u^{\frac{1}{2}}$.

Step2: Apply the chain rule

The chain rule states that if $y = u^{\frac{1}{2}}$ and $u=x^{3}-x^{2}+5x$, then $y'=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$: $\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}$. Then find $\frac{du}{dx}$: $\frac{du}{dx}=3x^{2}-2x + 5$.

Step3: Substitute $u$ back in

Substitute $u=x^{3}-x^{2}+5x$ into the derivative formula. So $y'=\frac{1}{2}(x^{3}-x^{2}+5x)^{-\frac{1}{2}}(3x^{2}-2x + 5)$.

Answer:

$y'=\frac{3x^{2}-2x + 5}{2\sqrt{x^{3}-x^{2}+5x}}$