Find the Derivative - d/dx y=(e^x)/(2x+1)

Problem

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

Solution

1. Identify the rule needed for the derivative. Since the function is a quotient of two expressions, u=e^x and v=2*x+1 use the quotient rule.

2. Apply the quotient rule formula, which is (vd(u)/d(x)−ud(v)/d(x))/(v^2)

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

4. Differentiate the denominator v=2*x+1 to get d(2*x+1)/d(x)=2

5. Substitute these derivatives into the quotient rule formula.

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

6. Factor out the common term e^x in the numerator to simplify the expression.

d(y)/d(x)=(e^x*(2*x+1−2))/((2*x+1)^2)

7. Simplify the terms inside the parentheses.

d(y)/d(x)=(e^x*(2*x−1))/((2*x+1)^2)

Final Answer

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

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