How do you differentiate f(x)=4x ln(3sin^2x^2 + 2) using the chain rule?

Nov 6, 2015

$f ' \left(x\right) = 4 \ln \left(3 {\sin}^{2} {x}^{2} + 2\right) + \frac{48 {x}^{2} \cos {x}^{2}}{3 {\sin}^{2} {x}^{2} + 2}$

Explanation:

$f ' \left(x\right) = \left(4 x\right) ' \cdot \ln \left(3 {\sin}^{2} {x}^{2} + 2\right) + 4 x \cdot \left(\ln \left(3 {\sin}^{2} {x}^{2} + 2\right)\right) '$

$f ' \left(x\right) = 4 \ln \left(3 {\sin}^{2} {x}^{2} + 2\right) + 4 x \cdot \frac{1}{3 {\sin}^{2} {x}^{2} + 2} \cdot \left(3 {\sin}^{2} {x}^{2} + 2\right) '$

$f ' \left(x\right) = 4 \ln \left(3 {\sin}^{2} {x}^{2} + 2\right) + \frac{4 x}{3 {\sin}^{2} {x}^{2} + 2} \cdot \left(3 {\sin}^{2} {x}^{2}\right) ' + 0$

$f ' \left(x\right) = 4 \ln \left(3 {\sin}^{2} {x}^{2} + 2\right) + \frac{4 x}{3 {\sin}^{2} {x}^{2} + 2} \cdot 6 \left(\sin {x}^{2}\right) '$

$f ' \left(x\right) = 4 \ln \left(3 {\sin}^{2} {x}^{2} + 2\right) + \frac{4 x}{3 {\sin}^{2} {x}^{2} + 2} \cdot 6 \cos {x}^{2} \cdot \left({x}^{2}\right) '$

$f ' \left(x\right) = 4 \ln \left(3 {\sin}^{2} {x}^{2} + 2\right) + \frac{4 x}{3 {\sin}^{2} {x}^{2} + 2} \cdot 6 \cos {x}^{2} \cdot 2 x$

$f ' \left(x\right) = 4 \ln \left(3 {\sin}^{2} {x}^{2} + 2\right) + \frac{48 {x}^{2} \cos {x}^{2}}{3 {\sin}^{2} {x}^{2} + 2}$