Find the Derivative - d/dx (9(x^2-3))/(x^3)

Problem

d()/d(x)(9*(x2−3))/(x3)

Solution

1. Simplify the expression before differentiating by distributing the constant and splitting the fraction into two separate terms.

(9*x2−27)/(x3)=(9*x2)/(x3)−27/(x3)

2. Rewrite the terms using negative exponents to prepare for the power rule.

9*x(−1)−27*x(−3)

3. Apply the power rule, which states that d(xn)/d(x)=n*x(n−1) to each term individually.

d()/d(x)*9*x(−1)=−9*x(−2)

d()/d(x)−27*x(−3)=81*x(−4)

4. Combine the results and rewrite the expression with positive exponents.

−9*x(−2)+81*x(−4)=−9/(x2)+81/(x4)

5. Find a common denominator to express the derivative as a single fraction.

−(9*x2)/(x4)+81/(x4)=(81−9*x2)/(x4)

Final Answer

d()/d(x)(9*(x2−3))/(x3)=(81−9*x2)/(x4)

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