Differentiation Rules Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
Differentiate f(x)=exx2+1f(x) = \dfrac{e^x}{x^2 + 1} using the quotient rule.

Solution

  1. 1
    Numerator u=exu = e^x, denominator v=x2+1v = x^2+1; derivatives u′=exu' = e^x, v′=2xv' = 2x.
  2. 2
    Quotient rule: f′(x)=ex(x2+1)−ex(2x)(x2+1)2f'(x) = \frac{e^x(x^2+1) - e^x(2x)}{(x^2+1)^2}.
  3. 3
    Factor exe^x from the numerator: f′(x)=ex(x2−2x+1)(x2+1)2=ex(x−1)2(x2+1)2f'(x) = \frac{e^x(x^2 - 2x + 1)}{(x^2+1)^2} = \frac{e^x(x-1)^2}{(x^2+1)^2}.

Answer

f′(x)=ex(x−1)2(x2+1)2f'(x) = \frac{e^x(x-1)^2}{(x^2+1)^2}
After applying the quotient rule, factoring exe^x from the numerator and recognising x2−2x+1=(x−1)2x^2-2x+1=(x-1)^2 simplifies the answer significantly.

About Differentiation Rules

A set of standard formulas for finding derivatives of common function types without using the limit definition each time.

Learn more about Differentiation Rules →

More Differentiation Rules Examples

Example 1 easy

Use the product rule to differentiate [formula].

Example 2 medium

Use the quotient rule to differentiate [formula].

Example 3 easy

Differentiate [formula] using the product rule.

← All Differentiation Rules Examples Differentiation Rules Concept