Find the Derivative - d/dx square root of 25-x^2

Problem

d(√(,25−x2))/d(x)

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, which states that the derivative of ƒ*(g(x)) is ƒ′*(g(x))⋅g(x)′

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 inner derivative 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.

(−2*x)/(2√(,25−x2))

6. Reduce the fraction to its simplest form.

(−x)/√(,25−x2)

Final Answer

d(√(,25−x2))/d(x)=−x/√(,25−x2)

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