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

Problem

d()/d(x)*x√(,16−x2)

Solution

1. Identify the rule needed for the expression, which is a product of x and √(,16−x2) We must use the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Rewrite the square root as a power to make differentiation easier.

√(,16−x2)=(16−x2)(1/2)

3. Apply the product rule where u=x and v=(16−x2)(1/2)

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

4. Differentiate the second term using the chain rule: (d(ƒ)*(g(x)))/d(x)=ƒ′*(g(x))⋅g(x)′

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

5. Simplify the derivative of the second term.

d(16−x2)/d(x)=−x*(16−x2)(−1/2)

6. Substitute the results back into the product rule equation.

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

7. Combine and simplify the expression.

d(y)/d(x)=(−x2)/√(,16−x2)+√(,16−x2)

8. Find a common denominator to combine the terms.

d(y)/d(x)=(−x2+(16−x2))/√(,16−x2)

9. Simplify the numerator.

d(y)/d(x)=(16−2*x2)/√(,16−x2)

Final Answer

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

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