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

Problem

d()/d(x)1/2*x2√(,16−x2)

Solution

1. Identify the function as a product of two terms, u=1/2*x2 and v=(16−x2)(1/2) which requires the product rule d()/d(x)*u*v=ud(v)/d(x)+vd(u)/d(x)

2. Differentiate the first term u=1/2*x2 using the power rule.

(d(1/2)*x2)/d(x)=x

3. Differentiate the second term v=(16−x2)(1/2) using the chain rule.

d(16−x2)/d(x)=1/2*(16−x2)(−1/2)*(−2*x)

d(16−x2)/d(x)=(−x)/√(,16−x2)

4. Apply the product rule by combining the derivatives.

d()/d(x)1/2*x2√(,16−x2)=1/2*x2*((−x)/√(,16−x2))+√(,16−x2)*(x)

5. Simplify the expression by finding a common denominator.

d()/d(x)1/2*x2√(,16−x2)=(−x3)/(2√(,16−x2))+(2*x*(16−x2))/(2√(,16−x2))

d()/d(x)1/2*x2√(,16−x2)=(−x3+32*x−2*x3)/(2√(,16−x2))

d()/d(x)1/2*x2√(,16−x2)=(32*x−3*x3)/(2√(,16−x2))

Final Answer

d()/d(x)1/2*x2√(,16−x2)=(32*x−3*x3)/(2√(,16−x2))

Want more problems? Check here!