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

Problem

d()/d(x)*x*(3*x−9)3

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x and v=(3*x−9)3

2. Apply the product rule, which states d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part, where d(x)/d(x)=1

4. Apply the chain rule to differentiate (3*x−9)3 which results in 3*(3*x−9)2⋅d()/d(x)*(3*x−9)

5. Calculate the derivative of the inner function d()/d(x)*(3*x−9)=3

6. Combine the results of the chain rule to find d(v)/d(x)=3*(3*x−9)2⋅3=9*(3*x−9)2

7. Substitute these components back into the product rule formula: x⋅9*(3*x−9)2+(3*x−9)3⋅1

8. Factor out the greatest common factor, which is (3*x−9)2

9. Simplify the remaining expression inside the brackets: (3*x−9)2*[9*x+(3*x−9)]

10. Combine like terms to get (3*x−9)2*(12*x−9)

11. Factor out a 3 from the second binomial to reach the final simplified form.

Final Answer

d()/d(x)*x*(3*x−9)3=3*(3*x−9)2*(4*x−3)

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