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

Problem

(d(x)√(,1−x^2))/d(x)

Solution

1. Identify the rule needed for the expression x√(,1−x^2) which is the product rule: (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

2. Assign the parts of the product where u=x and v=√(,1−x^2)=(1−x^2)^(1/2)

3. Differentiate u to find d(u)/d(x)=1

4. Differentiate v using the chain rule to find d(v)/d(x)=1/2*(1−x^2)^(−1/2)⋅(−2*x)=(−x)/√(,1−x^2)

5. Substitute these components into the product rule formula: x⋅(−x)/√(,1−x^2)+√(,1−x^2)⋅1

6. Simplify the expression by finding a common denominator: (−x^2)/√(,1−x^2)+(1−x^2)/√(,1−x^2)

7. Combine the terms to reach the final derivative: (1−2*x^2)/√(,1−x^2)

Final Answer

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

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