Find the Derivative - d/dx y=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 for the two terms x and (x−4)3

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

3. Differentiate the first part by setting u=x so d(x)/d(x)=1

4. Differentiate the second part by setting v=(x−4)3 and using the chain rule, resulting in d(x−4)/d(x)=3*(x−4)2⋅1

5. Combine the parts into the product rule formula.

d(y)/d(x)=x⋅3*(x−4)2+(x−4)3⋅1

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

d(y)/d(x)=(x−4)2*(3*x+(x−4))

7. Simplify the expression inside the parentheses.

d(y)/d(x)=(x−4)2*(4*x−4)

8. Factor out the constant 4 to reach the final simplified form.

d(y)/d(x)=4*(x−1)*(x−4)2

Final Answer

d()/d(x)*x*(x−4)3=4*(x−1)*(x−4)2

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