Question

this is section 2.8 problem 40: for the function y = 3xln(x), find $\frac{d^{2}y}{dx^{2}}$. answer: $\frac{d^{2}y}{dx^{2}}=$. hint: simplify before taking the second derivative

Explanation:

Step1: Find the first - derivative

Use the product rule $(uv)^\prime = u^\prime v+uv^\prime$, where $u = 3x$ and $v=\ln(x)$. The derivative of $u = 3x$ is $u^\prime=3$, and the derivative of $v=\ln(x)$ is $v^\prime=\frac{1}{x}$. Then $y^\prime=\frac{d}{dx}(3x\ln(x))=3\ln(x)+3x\cdot\frac{1}{x}=3\ln(x) + 3$.

Step2: Find the second - derivative

The derivative of $3\ln(x)$ is $\frac{3}{x}$ and the derivative of the constant 3 is 0. So $\frac{d^{2}y}{dx^{2}}=\frac{d}{dx}(3\ln(x)+3)=\frac{3}{x}$.

Answer:

$\frac{3}{x}$