Question

Consider the functionf (x) = x · In (x2)Find f"(2)

Answer 1

Derivatives

We are given the function

$f(x)=x\cdot\ln (x^2)$

We must find

$f^{\doubleprime}(2)$

Or the second derivative of f evaluated at x=2.

Find the first derivative. We must recall the following rules of derivatives:

$\begin{gathered} \lbrack f\cdot g\rbrack^{\prime}=f^{\prime}\cdot g+f\cdot g^{\prime} \\ (\ln u)^{\prime}=\frac{u^{\prime}}{u} \\ (x^n)^{\prime}=n\cdot x^{n-1} \end{gathered}$

Let's compute f'(x):compute

$\begin{gathered} f^{\prime}(x)=(x)^{\prime}\cdot\ln x^2+x\cdot(\ln x^2)^{\prime} \\ f^{\prime}(x)=(1)\cdot\ln x^2+x\cdot\frac{2x}{x^2} \\ f^{\prime}(x)=\ln x^2+2 \end{gathered}$

Now take the derivative again:

$\begin{gathered} f^{\doubleprime}(x)=(\ln x^2+2)^{\prime} \\ f^{\doubleprime}(x)=\frac{2x}{x^2}+0=\frac{2}{x} \end{gathered}$

Evaluating at x=2:

$f^{\doubleprime}(2)=\frac{2}{2}=1$