Find the Derivative - d/dx (e^x)/(1+x)

Problem

d()/d(x)(ex)/(1+x)

Solution

1. Identify the rule needed for the derivative of a quotient of two functions, which is the quotient rule.

2. Apply the formula for the quotient rule, d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2) where u=ex and v=1+x

3. Differentiate the numerator u=ex to get d(ex)/d(x)=ex

4. Differentiate the denominator v=1+x to get d(1+x)/d(x)=1

5. Substitute these derivatives into the quotient rule formula:

((1+x)*ex−e(1)x)/((1+x)2)

6. Simplify the numerator by distributing ex and combining like terms:

(ex+x*ex−ex)/((1+x)2)

7. Cancel the ex terms in the numerator to reach the final simplified form:

(x*ex)/((1+x)2)

Final Answer

d()/d(x)(ex)/(1+x)=(x*ex)/((1+x)2)

Want more problems? Check here!