Finding Derivative Values

In Exercises 67–72, find the value of (f ∘ g)' at the given value of x.

f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1

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First, understand that you need to find the derivative of the composite function (f ∘ g)(x), which means you need to apply the chain rule. The chain rule states that (f ∘ g)'(x) = f'(g(x)) * g'(x).

Calculate g(x) at x = -1. Substitute x = -1 into g(x) = (1 / x²) - 1 to find g(-1).

Next, find g'(x). Differentiate g(x) = (1 / x²) - 1 with respect to x. Use the power rule and the derivative of a constant.

Now, find f'(u). First, differentiate f(u) = ((u - 1) / (u + 1))² with respect to u. Use the chain rule and the quotient rule to differentiate this expression.

Finally, substitute g(-1) into f'(u) to find f'(g(-1)), and multiply this by g'(-1) to find (f ∘ g)'(-1).

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주요 개념

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Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. If you have a function h(x) = f(g(x)), the derivative h'(x) is found by multiplying the derivative of the outer function f at g(x) by the derivative of the inner function g at x. This rule is essential for finding derivatives of nested functions like f ∘ g.

Composite Functions

Composite functions involve applying one function to the results of another, denoted as (f ∘ g)(x) = f(g(x)). Understanding how to work with composite functions is crucial for applying the chain rule effectively, as it requires recognizing the inner and outer functions and their respective derivatives.

Derivative of Rational Functions

Rational functions are quotients of polynomials, and their derivatives can be found using the quotient rule. For a function f(u) = (u - 1)/(u + 1), the derivative involves differentiating the numerator and denominator separately and applying the quotient rule: (v'u - uv')/v², where u and v are the numerator and denominator, respectively.

Finding Derivative ValuesIn Exercises 67–72, find the value of (f ∘ g)' at the given value of x.f(u) = ((u − 1) / (u + 1))², u = g(x) = (1 / x²) − 1, x = −1