Find the Derivative - d/d@VAR f(x) = square root of 25-x^2

Problem

d()/d(x)√(,25−x2)

Solution

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

√(,25−x2)=(25−x2)(1/2)

2. Apply the chain rule by differentiating the outer function (the power) while keeping the inner function 25−x2 the same.

d(25−x2)/d(x)=1/2*(25−x2)(−1/2)⋅d(25−x2)/d(x)

3. Differentiate the inner function 25−x2 with respect to x

d(25−x2)/d(x)=−2*x

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

1/2*(25−x2)(−1/2)⋅(−2*x)

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

(−x)/((25−x2)(1/2))

6. Convert the fractional power back into radical form.

(−x)/√(,25−x2)

Final Answer

d(√(,25−x2))/d(x)=−x/√(,25−x2)

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