Find the Derivative - d/dx (4x-x^2)^3

Problem

d(4*x−x2)/d(x)

Solution

1. Identify the outer function and the inner function to apply the Chain Rule. The outer function is u3 and the inner function is u=4*x−x2

2. Apply the Power Rule to the outer function by bringing the exponent to the front and decreasing the exponent by one.

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

3. Differentiate the inner function 4*x−x2 using the Power Rule for each term.

d(4*x−x2)/d(x)=4−2*x

4. Substitute the derivative of the inner function back into the expression.

3*(4*x−x2)2*(4−2*x)

5. Simplify the expression by factoring a 2 out of the term (4−2*x) and multiplying it by the constant 3

3*(4*x−x2)2⋅2*(2−x)

6*(4*x−x2)2*(2−x)

Final Answer

d(4*x−x2)/d(x)=6*(4*x−x2)2*(2−x)

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