Find the Derivative - d/dx y=( natural log of x)/(e^(2x))

Problem

d()/d(x)ln(x)/(e(2*x))

Solution

1. Identify the function as a quotient of two functions, u=ln(x) and v=e(2*x) which requires the quotient rule.

2. Recall the quotient rule formula for a derivative.

d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

3. Differentiate the numerator u=ln(x) with respect to x

d(ln(x))/d(x)=1/x

4. Differentiate the denominator v=e(2*x) using the chain rule.

d(e(2*x))/d(x)=2*e(2*x)

5. Substitute these components into the quotient rule formula.

d(y)/d(x)=(e(2*x)⋅1/x−ln(x)⋅2*e(2*x))/((e(2*x))2)

6. Factor out the common term e(2*x) from the numerator to simplify the expression.

d(y)/d(x)=(e(2*x)*(1/x−2*ln(x)))/(e(4*x))

7. Simplify the fraction by dividing both the numerator and denominator by e(2*x)

d(y)/d(x)=(1/x−2*ln(x))/(e(2*x))

8. Clear the fraction in the numerator by multiplying the top and bottom by x

d(y)/d(x)=(1−2*x*ln(x))/(x*e(2*x))

Final Answer

d()/d(x)ln(x)/(e(2*x))=(1−2*x*ln(x))/(x*e(2*x))

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