Find the Derivative - d/dx y = square root of 9-x^2

Problem

d()/d(x)√(,9−x2)

Solution

1. Identify the function as a composition of functions, where the outer function is the square root u(1/2) and the inner function is u=9−x2

2. Apply the chain rule, which states that the derivative of ƒ*(g(x)) is ƒ′*(g(x))⋅g(x)′

3. Differentiate the outer function with respect to the inner function.

d(u(1/2))/d(u)=1/2*u(−1/2)

4. Differentiate the inner function with respect to x

d(9−x2)/d(x)=−2*x

5. Combine the results by multiplying the derivative of the outer function by the derivative of the inner function.

d(y)/d(x)=1/2*(9−x2)(−1/2)⋅(−2*x)

6. Simplify the expression by canceling the constant factors and moving the negative exponent to the denominator.

d(y)/d(x)=(−x)/√(,9−x2)

Final Answer

d(√(,9−x2))/d(x)=−x/√(,9−x2)

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