Question

find $\frac{d}{dx}left(\frac{x^{9}}{27}
ight)$. $\frac{d}{dx}left(\frac{x^{9}}{27}
ight)=square$

Explanation:

Step1: Recall the 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 \(\frac{dy}{dx}=c\cdot\frac{df(x)}{dx}\). Here \(c = \frac{1}{27}\) and \(f(x)=x^{9}\).

Step2: Recall the power - rule of differentiation

The power - rule states that if \(y = x^{n}\), then \(\frac{dy}{dx}=nx^{n - 1}\). For \(f(x)=x^{9}\), \(\frac{d}{dx}(x^{9})=9x^{8}\).

Step3: Apply the constant - multiple rule

Since \(\frac{d}{dx}(\frac{x^{9}}{27})=\frac{1}{27}\cdot\frac{d}{dx}(x^{9})\), substituting \(\frac{d}{dx}(x^{9}) = 9x^{8}\) into the equation, we get \(\frac{1}{27}\times9x^{8}\).

Step4: Simplify the expression

\(\frac{1}{27}\times9x^{8}=\frac{9x^{8}}{27}=\frac{1}{3}x^{8}\).

Answer:

\(\frac{1}{3}x^{8}\)