Determine h^(')(1) if h(x)=f(sqrt(g(2-x^(2))):} if g(1)=4,f^(')(2)=2,

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Determine  h^(')(1) if  h(x)=f(sqrt(g(2-x^(2))):} if  g(1)=4,f^(')(2)=2,

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Understand Functions Composition: Understand the composition of functions involved. We have h(x) = f(√(g(2 - x²))). To find h'(1), we need to use the chain rule, which allows us to differentiate composite functions.Apply Chain Rule: Apply the chain rule to differentiate h(x). The chain rule states that if we have a composite function h(x) = f(g(x)), then h'(x) = f'(g(x)) * g'(x). In our case, we have an additional layer due to the square root, so we will need to apply the chain rule multiple times.Differentiate Inner Function: Differentiate the innermost function g(2 - x²) with respect to x. g'(2 - x²) = -2x * g'(2 - x²) because the derivative of 2 - x² with respect to x is -2x.Differentiate Square Root: Differentiate the square root function with respect to its argument. If we have √u, where u is a function of x, then the derivative with respect to x is (1/2) * u^(-1/2) * u'. In our case, u = g(2 - x²), so we need to multiply the derivative from Step 3 by (1/2) * (g(2 - x²))^(-1/2).Differentiate Outer Function: Differentiate the outermost function f with respect to its argument. We are given that f'(2) = 2, but we need to evaluate f' at the point √(g(2 - x²)). We will use this information later when we plug in the specific values.Combine Derivatives: Combine the derivatives using the chain rule. h'(x) = f'(√(g(2 - x²))) * (1/2) * (g(2 - x²))^(-1/2) * -2x * g'(2 - x²)Evaluate at x = 1: Evaluate the derivative at x = 1. We need to find the values of g(2 - 1²), g'(2 - 1²), and f'(√(g(2 - 1²))) to compute h'(1).Calculate g(2 - 1²): Calculate g(2 - 1²) and g'(2 - 1²). Since g(1) = 4, we have g(2 - 1²) = g(1) = 4. We are not given g'(2 - 1²), but we can infer that g'(2 - 1²) = g'(1) since 2 - 1² = 1. However, we do not have the value of g'(1), so we cannot proceed further with the calculation.Recognize Missing Information: Recognize the missing information and conclude the problem cannot be solved with the given data. We cannot find h'(1) without knowing g'(1). Therefore, we cannot provide a numerical answer to the question prompt.

Determine $h'(1)$ if $h(x)=f(\sqrt{g(2-x^{2})})$ : if $g(1)=4$ , $f'(2)=2$ ,

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Determine h′(1)h'(1)h′(1) if h(x)=f(g(2−x2))h(x)=f(\sqrt{g(2-x^{2})})h(x)=f(g(2−x2)​): if g(1)=4g(1)=4g(1)=4, f′(2)=2f'(2)=2f′(2)=2,

Determine $h'(1)$ if $h(x)=f(\sqrt{g(2-x^{2})})$ : if $g(1)=4$ , $f'(2)=2$ ,