Question

How do you differentiate `g(x)= 3tan4x *sin2x*cos2x` using the product rule?

Answer 1

`(12sin(2x)cos(2x))/(cos^2(4x))+6tan(4x)*cos^2(2x)-6tan(4x)sin^2(2x)`

Explanation:

The rule is very simple, you derive one function and leave the others untouched, and then sum all these terms. So:

• Derive the first function and leave the other two untouched:

`(d/dx 3tan(4x))*sin(2x)*cos(2x)`

`= (3 * 1/(cos^2(4x))*4)*sin(2x)*cos(2x)`

• Derive the second function and leave the other two untouched:

`3tan(4x)* d/dx sin(2x)*cos(2x)`

`= 3tan(4x)* (cos(2x)* 2)* cos(2x)`

• Derive the third function and leave the other two untouched:

`3tan(4x)* sin(2x)* d/dx cos(2x)`

`= 3tan(4x)* sin(2x)* (-sin(2x)*2)`

We can readjust the three terms:

1. `(3 * 1/(cos^2(4x))*4)*sin(2x)*cos(2x) = (12sin(2x)cos(2x))/(cos^2(4x))`

2. `3tan(4x)* (cos(2x)* 2)* cos(2x) = 6tan(4x)*cos^2(2x)`

3. `3tan(4x)* sin(2x)* (-sin(2x)*2) = -6tan(4x)sin^2(2x)`

The result will be the sum of the three terms.