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

Problem

d()/d(x)*(−9/((3*x^2)^3))

Solution

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

(3*x^2)^3=3^3*(x^2)^3=27*x^6

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

y=−9/(27*x^6)=−1/(3*x^6)

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(x^n)/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/(x^7)

Final Answer

d()/d(x)*(−9/((3*x^2)^3))=2/(x^7)

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