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