\[ f(x) = \frac{e^{2x} \cdot \sin(3x)}{\sqrt{1 + x^2}} \]

Question

Find the derivative of the following function with respect to x:

\[ f(x) = \frac{e^{2x} \cdot \sin(3x)}{\sqrt{1 + x^2}} \]

Answer 1

Taking log both sides

log ( f(x) ) =  log ( e^2x ) + log ( sin(3x)) - ½ log ( 1 + x² )

log ( f(x) ) = 2x  + log ( sin(3x)) - ½ log ( 1 + x² )

Differentiating w.r.to x

1/f(x) . f’(x) =  2 + 3 cot(3x) - ½ (2x)/(1+ x²)

. f’(x) =   [ 2 + 3 cot(3x) -  x/(1+ x²) ]  (e^2x sin (3x))/sqrt( 1 + x² )