We have the expression: tan( ln (sqrt (4e^x+2x^2)))^2 + cot (ln (sqrt (4e^x + 2x^2)))^2

We have to find the derivative of the expression. We use the chain rule and start from the innermost function.

f(x) = tan( ln (sqrt (4e^x+2x^2)))^2 + cot (ln (sqrt (4e^x + 2x^2)))^2

f'(x) = 2*tan( ln (sqrt (4e^x+2x^2)))* (sec(ln (sqrt (4e^x+2x^2))))^2*(1/sqrt (4e^x+2x^2))*(1/2)*(1/sqrt (4e^x+2x^2))*(4e^x + 4x) + 2*cot (ln (sqrt (4e^x + 2x^2)))*(cosec(ln (sqrt (4e^x+2x^2))))^2*(1/sqrt (4e^x+2x^2))*(1/2)*(1/sqrt (4e^x+2x^2))*(4e^x + 4x)

=> [2*tan( ln (sqrt (4e^x+2x^2)))* (sec(ln (sqrt (4e^x+2x^2))))^2* + 2*cot (ln (sqrt (4e^x + 2x^2)))*(cosec(ln (sqrt (4e^x+2x^2))))^2]*(1/sqrt (4e^x+2x^2))*(1/2)*(1/sqrt (4e^x+2x^2))*(4e^x + 4x)

=> [2*tan( ln (sqrt (4e^x+2x^2)))* (sec(ln (sqrt (4e^x+2x^2))))^2* + 2*cot (ln (sqrt (4e^x + 2x^2)))*(cosec(ln (sqrt (4e^x+2x^2))))^2]*(1/(4e^x+2x^2))*(1/2)*(4e^x + 4x)

**The required derivative is [2*tan( ln (sqrt (4e^x+2x^2)))* (sec(ln (sqrt (4e^x+2x^2))))^2* + 2*cot (ln (sqrt (4e^x + 2x^2)))*(cosec(ln (sqrt (4e^x+2x^2))))^2]*(1/(4e^x+2x^2))*(1/2)*(4e^x + 4x)**