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.

b. ƒ(x)g²(x), x = 0

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To find the derivative of the function ƒ(x)g²(x) at x = 0, we need to apply the product rule and the chain rule. The product rule states that the derivative of a product of two functions u(x) and v(x) is given by u'(x)v(x) + u(x)v'(x).

In this case, let u(x) = ƒ(x) and v(x) = g²(x). We need to find the derivative of v(x) using the chain rule. The chain rule states that the derivative of a composite function h(g(x)) is h'(g(x))g'(x).

Since v(x) = g²(x), we can express it as h(g(x)) where h(x) = x². The derivative h'(x) is 2x, so the derivative of v(x) = g²(x) is 2g(x)g'(x).

Now, apply the product rule: the derivative of ƒ(x)g²(x) is ƒ'(x)g²(x) + ƒ(x)(2g(x)g'(x)).

Substitute the given values at x = 0: ƒ'(0) = -3, g(0) = 1, g'(0) = 1/2, and ƒ(0) = 1. Calculate ƒ'(0)g²(0) + ƒ(0)(2g(0)g'(0)) using these values.

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주요 개념

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

Product Rule

The Product Rule is a fundamental principle in calculus used to differentiate products of two functions. It states that if you have two functions, ƒ(x) and g(x), the derivative of their product is given by ƒ'(x)g(x) + ƒ(x)g'(x). This rule is essential for finding the derivative of combinations of functions, such as ƒ(x)g²(x), as it allows us to apply the differentiation process systematically.

Chain Rule

The Chain Rule is another critical differentiation technique used when dealing with composite functions. It states that if a function y is composed of another function u, then the derivative of y with respect to x is the derivative of y with respect to u multiplied by the derivative of u with respect to x. In the context of g²(x), the Chain Rule helps us differentiate the square of the function g(x) effectively.

Evaluating Derivatives at Specific Points

Evaluating derivatives at specific points involves substituting the given x-value into the derivative expression to find the slope of the function at that point. In this problem, we need to compute the derivatives of ƒ(x)g²(x) at x = 0, which requires using the values of ƒ(0), g(0), ƒ'(0), and g'(0) provided in the table. This step is crucial for obtaining the final numerical result.

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.b. ƒ(x)g²(x), x = 0