Answer to Question #234833 in Calculus for Jese Junior

Question

Differentiate the following with respect to x.

b) f(x) = ln 2x . sin3 (x2 – 3)

Answer 1

Let us differentiate the function $f(x) = \ln 2x \cdot \sin 3 (x^2 – 3)$ with respect to x using the product rule $(h(x)g(x))'=h'(x)g(x)+h(x)g'(x)$ and the chain rule $h(g(x))'=h'(g(x))\cdot g'(x):$

$f'(x) = \frac{1}{2x}\cdot 2 \cdot \sin 3 (x^2 – 3)+\ln 2x \cdot \cos 3 (x^2 – 3)\cdot 3\cdot 2x\\ =\frac{ \sin 3 (x^2 – 3)}{x}+6x\cdot\ln 2x \cdot \cos 3 (x^2 – 3).$