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

Problem

d()/d(x)√(,1−x2)

Solution

1. Rewrite the square root as a power to prepare for the chain rule.

√(,1−x2)=(1−x2)(1/2)

2. Apply the power rule to the outer function, which is the exponent 1/2

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

3. Apply the chain rule by differentiating the inner function 1−x2

d(1−x2)/d(x)=−2*x

4. Substitute the derivative of the inner function back into the expression.

d(y)/d(x)=1/2*(1−x2)(−1/2)⋅(−2*x)

5. Simplify the expression by canceling the constants and moving the negative exponent to the denominator.

d(y)/d(x)=(−x)/((1−x2)(1/2))

6. Convert the fractional exponent back into radical form.

d(y)/d(x)=(−x)/√(,1−x2)

Final Answer

d(√(,1−x2))/d(x)=−x/√(,1−x2)

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