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

Problem

d()/d(x)*(−9/((3*x2)3))

Solution

1. Simplify the denominator by applying the power of a product rule (a*b)n=an*bn

(3*x2)3=3*(x2)3=27*x6

2. Rewrite the original function by substituting the simplified denominator and reducing the fraction.

y=−9/(27*x6)=−1/(3*x6)

3. Express the function using a negative exponent to prepare for the power rule.

y=−1/3*x(−6)

4. Apply the power rule for derivatives, which states d(xn)/d(x)=n*x(n−1)

d(y)/d(x)=−1/3⋅(−6)*x(−6−1)

5. Simplify the coefficients and the exponent.

d(y)/d(x)=2*x(−7)

6. Rewrite the final result using a positive exponent.

d(y)/d(x)=2/(x7)

Final Answer

d()/d(x)*(−9/((3*x2)3))=2/(x7)

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