Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.

x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4

Find the first derivatives of the following combinations at the given value of x.

d. ƒ(g(x)), x = 0

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To find the derivative of the composition of functions ƒ(g(x)) at x = 0, we use the chain rule. The chain rule states that the derivative of a composition of functions is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

The chain rule formula for the derivative of ƒ(g(x)) is: (ƒ(g(x)))' = ƒ'(g(x)) * g'(x).

First, evaluate g(x) at x = 0. From the given data, g(0) = 1.

Next, find ƒ'(g(x)) at x = 0. Since g(0) = 1, we need ƒ'(1). From the table, ƒ'(1) = 1/2.

Finally, find g'(x) at x = 0. From the table, g'(0) = 1/2. Now, apply the chain rule: (ƒ(g(x)))' at x = 0 is ƒ'(1) * g'(0) = (1/2) * (1/2).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Chain Rule

The Chain Rule is a fundamental theorem in calculus used to differentiate composite functions. It states that if you have a function ƒ(g(x)), the derivative can be found by multiplying the derivative of the outer function ƒ with the derivative of the inner function g. This rule is essential for solving problems involving nested functions, as it allows for the systematic calculation of derivatives.

Function Values and Derivatives

Understanding function values and their derivatives at specific points is crucial for applying calculus concepts. In this context, we need to evaluate ƒ(g(0)) and then find the derivative of that composition. The values of ƒ and g at x = 0, along with their derivatives, provide the necessary information to compute the derivative of the composite function using the Chain Rule.

Evaluating Derivatives at Specific Points

Evaluating derivatives at specific points involves substituting the given x-value into the derivative expression. In this case, after applying the Chain Rule, we will need to evaluate the resulting expression at x = 0. This step is critical for obtaining the final numerical result, as it translates the abstract derivative into a concrete value that reflects the behavior of the function at that point.

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.x ƒ(x) g(x) ƒ'(x) g'(x)0 1 1 -3 1/21 3 5 1/2 -4Find the first derivatives of the following combinations at the given value of x.d. ƒ(g(x)), x = 0