Find the following derivative: d/dx ((xe^x)/sin(2x))

Question

i dont know how to deal with e and sine in the same equation.

Answer 1

We will use the a combination of product, quotient, and chain rule. 

 

Let f(x) = xe^x         and         g(x) =  sin(2x)

 

Product rule

 

d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

 

Quotient rule

 

d/dx[f(x) / g(x)] = [f'(x)g(x) - f(x)g'(x)] / g²(x)

 

Chain rule

 

derivative of the inside function times the derivative of the outside function evaluated insides

 

Note that the derivative of and exponential function is itself.

 

Knowing this, we can derive the original function.

 

 

d/dx = [(e^x + xe^x)sin(2x) - (xe^x)2cos(2x)] / sin²(2x)

 

d/dx = [(e^x + xe^x)sin(2x) - 2xe^xcos(2x)] / sin²(2x)

 

d/dx = [e^xsin(2x) + xe^xsin(2x) - 2xe^xcos(2x)] / sin²(2x)

 

 

Now we can simplify the expression by dividing each term by sin²(2x).

 

d/dx = (e^x / sin(2x)) + (xe^x / sin(2x)) - (2xe^x cot(2x) / sin(2x))