Find the Derivative - d/dx x(6-2x)^2

Problem

d()/d(x)*x*(6−2*x)2

Solution

1. Identify the rule needed for the expression, which is the product rule (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) where u=x and v=(6−2*x)2

2. Apply the product rule to set up the derivative.

(d(x)*(6−2*x)2)/d(x)=xd(6−2*x)/d(x)+(6−2*x)2d(x)/d(x)

3. Apply the chain rule to differentiate (6−2*x)2 which gives 2*(6−2*x)⋅(−2)

d(6−2*x)/d(x)=2*(6−2*x)*(−2)

d(6−2*x)/d(x)=−4*(6−2*x)

4. Substitute the derivatives back into the product rule expression.

(d(x)*(6−2*x)2)/d(x)=x*(−4*(6−2*x))+(6−2*x)2*(1)

5. Simplify the expression by factoring out the common term (6−2*x)

(d(x)*(6−2*x)2)/d(x)=(6−2*x)*(−4*x+(6−2*x))

6. Combine like terms inside the second set of parentheses.

(d(x)*(6−2*x)2)/d(x)=(6−2*x)*(6−6*x)

7. Factor out constants to reach the simplest form.

(d(x)*(6−2*x)2)/d(x)=2*(3−x)⋅6*(1−x)

(d(x)*(6−2*x)2)/d(x)=12*(3−x)*(1−x)

Final Answer

d()/d(x)*x*(6−2*x)2=12*(3−x)*(1−x)

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