Find the Derivative - d/dx (e^x)/(x^3)

Problem

d()/d(x)(e^x)/(x^3)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, u=e^x and v=x^3 use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Differentiate the numerator and the denominator separately.

d(e^x)/d(x)=e^x

d(x^3)/d(x)=3*x^2

3. Apply the quotient rule by substituting the functions and their derivatives into the formula.

d()/d(x)(e^x)/(x^3)=(x^3*e^x−e^x*(3*x^2))/((x^3)^2)

4. Simplify the expression by multiplying the powers in the denominator and factoring the numerator.

d()/d(x)(e^x)/(x^3)=(x^3*e^x−3*x^2*e^x)/(x^6)

5. Factor out the common term x^2*e^x from the numerator.

d()/d(x)(e^x)/(x^3)=(x^2*e^x*(x−3))/(x^6)

6. Reduce the fraction by dividing both the numerator and denominator by x^2

d()/d(x)(e^x)/(x^3)=(e^x*(x−3))/(x^4)

Final Answer

d()/d(x)(e^x)/(x^3)=(e^x*(x−3))/(x^4)

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