Find dy/dx y=x square root of 1-x^2

Problem

d()/d(x)*(x√(,1−x2))

Solution

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

2. Differentiate the first term u=x with respect to x to get d(u)/d(x)=1

3. Differentiate the second term v=(1−x2)(1/2) using the chain rule, resulting in d(v)/d(x)=1/2*(1−x2)(−1/2)⋅(−2*x)

4. Simplify the derivative of the second term to d(v)/d(x)=(−x)/√(,1−x2)

5. Substitute these components back into the product rule formula: d(y)/d(x)=x((−x)/√(,1−x2))+√(,1−x2)*(1)

6. Combine the terms over a common denominator √(,1−x2) to simplify the expression.

7. Simplify the numerator: (−x2+(1−x2))/√(,1−x2)=(1−2*x2)/√(,1−x2)

Final Answer

d(y)/d(x)=(1−2*x2)/√(,1−x2)

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