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

Problem

d()/d(x)(ex)/(x3)

Solution

1. Identify the rule needed for differentiation. Since the expression is a quotient of two functions, u=ex and v=x3 use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Differentiate the numerator and the denominator separately.

d(ex)/d(x)=ex

d(x3)/d(x)=3*x2

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

d()/d(x)(ex)/(x3)=(x3*ex−ex*(3*x2))/((x3)2)

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

d()/d(x)(ex)/(x3)=(x3*ex−3*x2*ex)/(x6)

5. Factor out the common term x2*ex from the numerator.

d()/d(x)(ex)/(x3)=(x2*ex*(x−3))/(x6)

6. Reduce the fraction by dividing both the numerator and denominator by x2

d()/d(x)(ex)/(x3)=(ex*(x−3))/(x4)

Final Answer

d()/d(x)(ex)/(x3)=(ex*(x−3))/(x4)

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