Derivative of the quotient eˣ over (x²+1)

Problem

Differentiate $f(x) = \dfrac{e^x}{x^2+1}$ (quotient rule).

Solution

With $\left(\dfrac{f}{g}\right)' = \dfrac{f'g - fg'}{g^2}$, setting $f=e^x$ and $g=x^2+1$: $f'(x) = \frac{e^x(x^2+1) - e^x\cdot 2x}{(x^2+1)^2} = \frac{e^x(x^2-2x+1)}{(x^2+1)^2} = \boxed{\frac{e^x(x-1)^2}{(x^2+1)^2}}.$

Links

Topics: Derivatives
Concepts: Derivative · Rules of differentiation
Skills: Differentiate
Exercise type: Derivative computation