How do you differentiate f(x)= e^(2x)* (x^2 - 4) *ln x using the product rule?

Jan 5, 2017

$f ' \left(x\right) = {e}^{2 x} \left[\ln x \left({x}^{2} - 4\right) + 2 x \ln x + \frac{1}{x} \left({x}^{2} - 4\right)\right]$

Explanation:

#To differentiate the product of 3 functions

$\left(a b c\right) ' = a ' b c + b ' a c + c ' a b$

$\Rightarrow a = {e}^{2 x} \Rightarrow a ' = {e}^{2 x} . \frac{d}{\mathrm{dx}} \left(2 x\right) = 2 {e}^{2 x}$

$b = {x}^{2} - 4 \Rightarrow b ' = 2 x$

$\text{and } c = \ln x \Rightarrow c ' = \frac{1}{x}$

$\Rightarrow f ' \left(x\right) = 2 {e}^{2 x} \left(\ln x \left({x}^{2} - 4\right)\right) + 2 x \left({e}^{2 x} \ln x\right) + \frac{1}{x} \left({e}^{2 x} \left({x}^{2} - 4\right)\right)$

$= {e}^{2 x} \left[\ln x \left({x}^{2} - 4\right) + 2 x \ln x + \frac{1}{x} \left({x}^{2} - 4\right)\right]$