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

Problem

d()/d(x)(e^x)/(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))/(v^2) where u=e^x and v=1+x

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

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)*e^x−e(1)^x)/((1+x)^2)

6. Simplify the numerator by distributing e^x and combining like terms:

(e^x+x*e^x−e^x)/((1+x)^2)

7. Cancel the e^x terms in the numerator to reach the final simplified form:

(x*e^x)/((1+x)^2)

Final Answer

d()/d(x)(e^x)/(1+x)=(x*e^x)/((1+x)^2)

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