find $\frac{d}{dx}(\frac{x^{8}}{24})$. $\frac{d}{dx}(\frac{x^{8}}{24})=square$

Explanation:

Step1: Use constant - multiple rule of differentiation

The constant - multiple rule states that if $y = c\cdot f(x)$ where $c$ is a constant and $f(x)$ is a function of $x$, then $\frac{d}{dx}(c\cdot f(x))=c\cdot\frac{d}{dx}(f(x))$. Here $c = \frac{1}{24}$ and $f(x)=x^{8}$. So, $\frac{d}{dx}\left(\frac{x^{8}}{24}\right)=\frac{1}{24}\cdot\frac{d}{dx}(x^{8})$.

Step2: Use power rule of differentiation

The power rule states that $\frac{d}{dx}(x^{n})=nx^{n - 1}$ for any real - number $n$. For $n = 8$, $\frac{d}{dx}(x^{8})=8x^{7}$.

Step3: Calculate the final result

Substitute $\frac{d}{dx}(x^{8}) = 8x^{7}$ into the equation from Step 1. We get $\frac{1}{24}\cdot8x^{7}=\frac{1}{3}x^{7}$.

Answer:

$\frac{1}{3}x^{7}$