Find the Derivative - d/dx x(12-2x)^2

Problem

d()/d(x)*x*(12−2*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=(12−2*x)2

2. Differentiate the first part u=x with respect to x

d(x)/d(x)=1

3. Differentiate the second part v=(12−2*x)2 using the chain rule.

d(12−2*x)/d(x)=2*(12−2*x)⋅(−2)

d(12−2*x)/d(x)=−4*(12−2*x)

4. Apply the product rule by combining the derivatives.

d()/d(x)*x*(12−2*x)2=x*(−4*(12−2*x))+(12−2*x)2*(1)

5. Factor out the common term (12−2*x) to simplify the expression.

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

6. Simplify the terms inside the second set of parentheses.

d()/d(x)*x*(12−2*x)2=(12−2*x)*(12−6*x)

7. Factor out constants to reach the simplest form.

d()/d(x)*x*(12−2*x)2=2*(6−x)⋅6*(2−x)

d()/d(x)*x*(12−2*x)2=12*(6−x)*(2−x)

Final Answer

d()/d(x)*x*(12−2*x)2=12*(6−x)*(2−x)

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