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Find the derivative of e^x - 1 / e² + 1

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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).

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