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

Feb 27, 2016

${e}^{{\left(\ln \left({x}^{2} + 3\right)\right)}^{2}} \cdot 4 x \left(\ln \frac{{x}^{2} + 3}{{x}^{2} + 3}\right)$

Explanation:

first find the derivative of $\left({\left(\ln \left({x}^{2} + 3\right)\right)}^{2}\right)$, the expression in the power of e then multiply it with ${e}^{{\left(\ln \left({x}^{2} + 3\right)\right)}^{2}}$

Set $u \left(x\right) = {\left(\ln \left({x}^{2} + 3\right)\right)}^{2}$ hence

$\left(\frac{\mathrm{df}}{\mathrm{dx}}\right) = {e}^{u \left(x\right)} \cdot \left(\frac{\mathrm{du}}{\mathrm{dx}}\right)$

where

$\frac{\mathrm{du}}{\mathrm{dx}} = \frac{4 x \ln \left({x}^{2} + 3\right)}{{x}^{2} + 3}$