Question

this is section 2.8 problem 28: for the function y = ln(\sqrt5{x^{3}}), find \frac{dy}{dx}. answer: \frac{dy}{dx}= . hint: follow example 6. simplify first!

Explanation:

Step1: Simplify the function

First, rewrite $\sqrt[5]{x^{3}}$ as $x^{\frac{3}{5}}$. So $y = \ln(x^{\frac{3}{5}})$. Using the property of logarithms $\ln(a^{b})=b\ln(a)$, we get $y=\frac{3}{5}\ln(x)$.

Step2: Differentiate the simplified - function

The derivative of $\ln(x)$ with respect to $x$ is $\frac{1}{x}$. Using the constant - multiple rule of differentiation $(cf(x))^\prime = cf^\prime(x)$ where $c=\frac{3}{5}$ and $f(x)=\ln(x)$. So $\frac{dy}{dx}=\frac{3}{5}\times\frac{1}{x}=\frac{3}{5x}$.

Answer:

$\frac{3}{5x}$