Question

Let f(x) = e^(3x sin (x)). find f'(x)

Answer 1

`f^(\prime)(x)=3(\sin x+x\cos x)e^(3x\sin x)`

Step-by-step explanation:

Given the function

`f(x)=e^(3x\sin x)`

f'(x) represents the first derivative of f(x) with respect to x.

This can be done by using the principle of function of a function.

Let u = 3x sinx

Then

f'(x) = f'(u).u'(x)

`\begin{gathered} f(u)=e^u \\ f^(\prime)(u)=e^u \end{gathered}`
`\begin{gathered} u(x)=3x\sin x \\ u^(\prime)(x)=3\sin x+3x\cos x \end{gathered}`

Therefore;

`f^(\prime)(x)=e^u.(3\sin x+3x\cos x)`

with u = 3x sinx, we have

`\begin{gathered} f^(\prime)(x)=(3\sin x+3x\cos x)e^(3x\sin x) \\ \\ =3(\sin x+x\cos x)e^(3x\sin x) \end{gathered}`