How do you find the second derivative of f(x)=ln(7x^2e^x\sin x)?

Mar 31, 2018

$\frac{{d}^{2} f}{{\mathrm{dx}}^{2}} = - \frac{2}{x} ^ 2 - {\csc}^{2} x$

Explanation:

As $f \left(x\right) = \ln \left(7 {x}^{2} {e}^{x} \sin x\right)$

i.e. $f \left(x\right) = \ln 7 + 2 \ln x + x + \ln \sin x$

Hence $\frac{\mathrm{df}}{\mathrm{dx}} = \frac{2}{x} + 1 + \frac{1}{\sin} x \cdot \cos x$

= $\frac{2}{x} + 1 + \cot x$

or $\frac{{d}^{2} f}{{\mathrm{dx}}^{2}} = - \frac{2}{x} ^ 2 - {\csc}^{2} x$