Answer:
Explanation:
To find the derivative of the given function f(x), we can use the product rule and the chain rule of differentiation:
f(x) = (1+x^2)e^(2x) + 5/(1+2x^2)
f'(x) = [(1+x^2)(d/dx[e^(2x)]) + (d/dx[1+x^2])(e^(2x))] + [(d/dx[5])(1+2x^2) - 5(d/dx[1+2x^2])(d/dx[1+2x^2])]/(1+2x^2)^2
Using the chain rule, we get:
d/dx[e^(2x)] = e^(2x) * d/dx[2x] = 2e^(2x)
Substituting this in the above expression, we get:
f'(x) = [(1+x^2)(2e^(2x)) + 2xe^(2x)] + [(0)(1+2x^2) - 5(2x)(0)]/(1+2x^2)^2
Simplifying further, we get:
f'(x) = 2e^(2x) + 2xe^(2x) + (-10x)/(1+2x^2)^2
Therefore, the derivative of f(x) is:
f'(x) = 2e^(2x) + 2xe^(2x) - (10x)/(1+2x^2)^2