Find the Derivative - d/dx (x-4)^2(4x-4)

Problem

d()/d(x)*(x−4)2*(4*x−4)

Solution

1. Identify the product rule, which states that d()/d(x)*ƒ(x)*g(x)=ƒ(x)d(g(x))/d(x)+g(x)d(ƒ(x))/d(x)

2. Assign the functions ƒ(x)=(x−4)2 and g(x)=4*x−4

3. Differentiate ƒ(x) using the power rule and chain rule to get d(x−4)/d(x)=2*(x−4)

4. Differentiate g(x) using the power rule to get d(4*x−4)/d(x)=4

5. Apply the product rule formula by substituting the functions and their derivatives.

d()/d(x)*(x−4)2*(4*x−4)=(x−4)2*(4)+(4*x−4)*(2*(x−4))

6. Factor out the common term 4*(x−4) from the expression.

4*(x−4)*[(x−4)+2*(x−1)]

7. Simplify the expression inside the brackets.

4*(x−4)*[x−4+2*x−2]

8. Combine like terms to reach the final simplified form.

4*(x−4)*(3*x−6)

9. Factor out a 3 from the second binomial to further simplify.

12*(x−4)*(x−2)

Final Answer

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

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