Find the Derivative - d/dx f(x)=x(4x-12)^3

Problem

d()/d(x)*x*(4*x−12)3

Solution

1. Identify the rule needed for the expression, which is a product of two functions: u=x and v=(4*x−12)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 u=x to get d(x)/d(x)=1

4. Differentiate the second part v=(4*x−12)3 using the chain rule, which gives 3*(4*x−12)2⋅d()/d(x)*(4*x−12)

5. Simplify the derivative of the second part to 3*(4*x−12)2⋅4=12*(4*x−12)2

6. Substitute these components back into the product rule formula:

x⋅12*(4*x−12)2+(4*x−12)3⋅1

7. Factor out the greatest common factor, which is (4*x−12)2

(4*x−12)2*[12*x+(4*x−12)]

8. Combine like terms inside the brackets:

(4*x−12)2*(16*x−12)

9. Factor out constants to simplify further if desired. From (4*x−12)2 factor out 4=16 From (16*x−12) factor out 4

16*(x−3)2⋅4*(4*x−3)

64*(x−3)2*(4*x−3)

Final Answer

(d(x)*(4*x−12)3)/d(x)=64*(x−3)2*(4*x−3)

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