(d)/(dx)×sqrt(100-x^(2))

Identify Functions: We are asked to find the derivative of the function f(x) = sqrt(100-x^2) with respect to x. We will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Find Derivatives: First, let's identify the outer function and the inner function. The outer function is g(u) = sqrt(u), and the inner function is u(x) = 100 - x^2. We will need to find g'(u) and u'(x).Apply Chain Rule: The derivative of the outer function g(u) = sqrt(u) with respect to u is g'(u) = 1/(2sqrt(u)).Calculate Derivative: The derivative of the inner function u(x) = 100 - x^2 with respect to x is u'(x) = -2x.Simplify Expression: Now, we apply the chain rule. The derivative of f(x) with respect to x is f'(x) = g'(u(x)) * u'(x). Substituting the derivatives we found, we get f'(x) = (1/(2sqrt(100-x^2))) * (-2x).Final Derivative: Simplify the expression by multiplying the two derivatives together. f'(x) = -2x / (2sqrt(100-x^2)).We can simplify the expression further by canceling out the 2 in the numerator and the denominator. f'(x) = -x / sqrt(100-x^2).

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$\frac{d}{d x} \times \sqrt{100-x^{2}}$

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Algebra 2

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\frac{d}{d x} \times \sqrt{100-x^{2}}

Identify Functions: We are asked to find the derivative of the function $f(x) = \sqrt{100-x^2}$ with respect to $x$ . We will use the chain rule, which states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Find Derivatives: First, let's identify the outer function and the inner function. The outer function is $g(u) = \sqrt{u}$ , and the inner function is $u(x) = 100 - x^2$ . We will need to find $g'(u)$ and $u'(x)$ .
Apply Chain Rule: The derivative of the outer function $g(u) = \sqrt{u}$ with respect to $u$ is $g'(u) = \frac{1}{2\sqrt{u}}$ .
Calculate Derivative: The derivative of the inner function $u(x) = 100 - x^2$ with respect to $x$ is $u'(x) = -2x$ .
Simplify Expression: Now, we apply the chain rule. The derivative of $f(x)$ with respect to $x$ is $f'(x) = g'(u(x)) \cdot u'(x)$ . Substituting the derivatives we found, we get $f'(x) = \frac{1}{2\sqrt{100-x^2}} \cdot (-2x)$ .
Final Derivative: Simplify the expression by multiplying the two derivatives together. $f'(x) = \frac{-2x}{2\sqrt{100-x^2}}$ .
Final Derivative: Simplify the expression by multiplying the two derivatives together. $f'(x) = -2x / (2\sqrt{100-x^2})$ .We can simplify the expression further by canceling out the $2$ in the numerator and the denominator. $f'(x) = -x / \sqrt{100-x^2}$ .