Find the Derivative Using Quotient Rule - d/dx 1/(x^3)

Problem

d()/d(x)1/(x3)

Solution

1. Identify the numerator u=1 and the denominator v=x3 to apply the quotient rule.

2. Find the derivatives of the numerator and denominator: d(u)/d(x)=0 and d(v)/d(x)=3*x2

3. Apply the quotient rule formula d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

4. Substitute the values into the formula:

((x3)*(0)−(1)*(3*x2))/((x3)2)

5. Simplify the numerator and the denominator:

(0−3*x2)/(x6)

6. Reduce the fraction by dividing both terms by x2

(−3*x2)/(x6)=−3/(x4)

Final Answer

d()/d(x)1/(x3)=−3/(x4)

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