Find the Derivative - d/dx 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(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the first part u=x to get d(x)/d(x)=1

4. Differentiate the second part v=(3*x−9)3 using the chain rule, which gives 3*(3*x−9)2⋅d(3*x−9)/d(x)=3*(3*x−9)2⋅3=9*(3*x−9)2

5. Substitute these derivatives back into the product rule formula:

x⋅9*(3*x−9)2+(3*x−9)3⋅1

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

(3*x−9)2*[9*x+(3*x−9)]

7. Simplify the expression inside the brackets:

(3*x−9)2*(12*x−9)

8. Factor out a 3 from the second term and a 3 from the first term to simplify further:

[3*(x−3)]2⋅3*(4*x−3)

9*(x−3)2⋅3*(4*x−3)

27*(x−3)2*(4*x−3)

Final Answer

d()/d(x)*x*(3*x−9)3=27*(x−3)2*(4*x−3)

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