What is the second derivative of the function f(x) given by f(x)=e^2x+sin2x/2x ?

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We have the function f(x) = e^2x + (sin 2x) / 2x

f'(x) = [e^2x + (sin 2x) / 2x]'

=> (e^2x)' + [(sin 2x/2x]'

=> 2*e^2x + (2x*2*cos 2x - 2*sin 2x )/4*x^2

=> 2*e^2x + (2x *cos 2x - sin 2x  )/2*x^2

f''(x) = (f'(x))'

=> [2*e^2x + (cos 2x) / x - (sin 2x)/2*x^2]'

=> 4*e^2x + ((cos 2x) / x)' - [(sin 2x)/2*x^2]'

=> 4*e^2x - (cos 2x + 2*x*sin 2x)/x^2 - (1/2)*[(sin 2x)*2x + 2*x^2* cos 2x)/x^4]

=> 4*e^2x + (cos 2x + 2*x*sin 2x)/x^2 - [(sin 2x + x*cos 2x)/x^3]

The required derivative is 4*e^2x + (cos 2x + 2*x*sin 2x)/x^2 - (sin 2x + x*cos 2x)/x^3

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