find the derivative of $y = \\left(\\frac{x^{2}+7}{8}\\right)^{8}$ $\\frac{dy}{dx}=$

Explanation:

Step1: Apply chain - rule

Let $u=\frac{x^{2}+7}{8}$, then $y = u^{8}$. The chain - rule states that $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. First, find $\frac{dy}{du}$. Using the power rule $\frac{d}{du}(u^{n})=nu^{n - 1}$, for $y = u^{8}$, we have $\frac{dy}{du}=8u^{7}$.

Step2: Find $\frac{du}{dx}$

Since $u=\frac{x^{2}+7}{8}=\frac{1}{8}x^{2}+\frac{7}{8}$, then $\frac{du}{dx}=\frac{1}{8}\cdot2x=\frac{x}{4}$.

Step3: Calculate $\frac{dy}{dx}$

Substitute $u=\frac{x^{2}+7}{8}$ back into $\frac{dy}{du}$ and multiply by $\frac{du}{dx}$. $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=8u^{7}\cdot\frac{x}{4}$. Replace $u$ with $\frac{x^{2}+7}{8}$: $\frac{dy}{dx}=8(\frac{x^{2}+7}{8})^{7}\cdot\frac{x}{4}=\frac{x(x^{2}+7)^{7}}{4}$.

Answer:

$\frac{x(x^{2}+7)^{7}}{4}$