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

Problem

d()/d(x)*x2*(x−2)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)=x2 and g(x)=(x−2)4

3. Differentiate ƒ(x) using the power rule to get d(x2)/d(x)=2*x

4. Differentiate g(x) using the chain rule to get d(x−2)/d(x)=4*(x−2)3⋅1=4*(x−2)3

5. Apply the product rule formula by substituting the derivatives back into the expression.

(d(x2)*(x−2)4)/d(x)=x2⋅4*(x−2)3+(x−2)4⋅2*x

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

(d(x2)*(x−2)4)/d(x)=2*x*(x−2)3*[2*x+(x−2)]

7. Simplify the expression inside the brackets.

(d(x2)*(x−2)4)/d(x)=2*x*(x−2)3*(3*x−2)

Final Answer

(d(x2)*(x−2)4)/d(x)=2*x*(x−2)3*(3*x−2)

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