Find the Derivative - d/dx 1/2*(x^2 square root of 16-x^2)

Problem

d()/d(x)1/2*x^2√(,16−x^2)

Solution

1. Identify the constant and the product of functions. Pull the constant 1/2 out of the derivative.

d()/d(x)1/2*x^2√(,16−x^2)=1/2(d(x^2)√(,16−x^2))/d(x)

2. Apply the product rule, which states (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x) where u=x^2 and v=√(,16−x^2)

1/2*(x^2d(√(,16−x^2))/d(x)+√(,16−x^2)d(x^2)/d(x))

3. Apply the chain rule to differentiate √(,16−x^2) treating it as (16−x^2)^(1/2)

d(√(,16−x^2))/d(x)=1/2*(16−x^2)^(−1/2)⋅(−2*x)

d(√(,16−x^2))/d(x)=(−x)/√(,16−x^2)

4. Differentiate the second part of the product, x^2 using the power rule.

d(x^2)/d(x)=2*x

5. Substitute these derivatives back into the product rule expression.

1/2*(x((−x)/√(,16−x^2))^2+√(,16−x^2)*(2*x))

6. Simplify the expression inside the parentheses by finding a common denominator.

1/2*((−x^3)/√(,16−x^2)+(2*x*(16−x^2))/√(,16−x^2))

1/2*((−x^3+32*x−2*x^3)/√(,16−x^2))

1/2*((32*x−3*x^3)/√(,16−x^2))

7. Distribute the 1/2 to reach the final simplified form.

(32*x−3*x^3)/(2√(,16−x^2))

Final Answer

d()/d(x)1/2*x^2√(,16−x^2)=(32*x−3*x^3)/(2√(,16−x^2))

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