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

Problem

d()/d(x)*x*(x−4)3

Solution

1. Identify the rule needed for the expression, which is the product rule d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x) where u=x and v=(x−4)3

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

3. Apply the chain rule to differentiate the second part v=(x−4)3 resulting in d(v)/d(x)=3*(x−4)2⋅d(x−4)/d(x)=3*(x−4)2

4. Substitute these components into the product rule formula.

(d(x)*(x−4)3)/d(x)=x⋅3*(x−4)2+(x−4)3⋅1

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

(d(x)*(x−4)3)/d(x)=(x−4)2*[3*x+(x−4)]

6. Simplify the expression inside the brackets by combining like terms.

(d(x)*(x−4)3)/d(x)=(x−4)2*(4*x−4)

7. Factor out the constant 4 from the linear term to reach the final simplified form.

(d(x)*(x−4)3)/d(x)=4*(x−1)*(x−4)2

Final Answer

(d(x)*(x−4)3)/d(x)=4*(x−1)*(x−4)2

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