Answer:
To find the derivative of the function f(x) = (e^x - 1) / (e^2 + 1), we can use the quotient rule of differentiation.
Let’s denote the numerator as u(x) = e^x - 1 and the denominator as v(x) = e^2 + 1.
The quotient rule states that if we have a function h(x) = u(x) / v(x), then the derivative of h(x) is given by:
h’(x) = (u’(x)v(x) - v’(x)u(x)) / (v(x))^2
Now, let’s find the derivatives of u(x) and v(x):
u’(x) = d/dx(e^x - 1) = e^x
v’(x) = d/dx(e^2 + 1) = 0 (since e^2 + 1 is a constant)
Using these derivatives, we can substitute into the quotient rule formula:
h’(x) = (e^x * (e^2 + 1) - 0 * (e^x - 1)) / (e^2 + 1)^2
Simplifying the numerator:
h’(x) = e^x * (e^2 + 1) / (e^2 + 1)^2
Finally, we can simplify further by canceling out the common factors:
h’(x) = e^x / (e^2 + 1)
Therefore, the derivative of f(x) = (e^x - 1) / (e^2 + 1) is e^x / (e^2 + 1).