Finding Derivative Values

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

f(u) = cot(πu/10), u = g(x) = 5√x, x = 1

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First, understand that you need to find the derivative of the composite function (f ∘ g)(x), which is f(g(x)). This involves using the chain rule for differentiation.

Identify the functions involved: f(u) = cot(πu/10) and u = g(x) = 5√x. You need to differentiate f with respect to u and g with respect to x.

Apply the chain rule: (f ∘ g)'(x) = f'(g(x)) * g'(x). This means you need to find f'(u) and g'(x) and then multiply them.

Differentiate f(u) = cot(πu/10) with respect to u. Use the derivative of cotangent, which is -csc²(u), and apply the chain rule to account for the π/10 factor.

Differentiate g(x) = 5√x with respect to x. Recall that √x can be expressed as x^(1/2), and use the power rule to find g'(x).

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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 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 with respect to its inner function g, by the derivative of the inner function g with respect to x. This is expressed as h'(x) = f'(g(x)) * g'(x).

Derivative of Cotangent Function

The derivative of the cotangent function, cot(u), with respect to u is -csc^2(u). This derivative is crucial when differentiating functions involving cotangent, as it allows us to apply the chain rule effectively. In the context of the given problem, understanding this derivative helps in finding the derivative of f(u) = cot(πu/10).

Square Root Function Derivative

The derivative of the square root function, √x, is (1/2)x^(-1/2). This derivative is essential when dealing with functions that involve square roots, such as g(x) = 5√x. Knowing this allows us to compute g'(x), which is necessary for applying the chain rule to find the derivative of the composite function (f ∘ g)(x).

Finding Derivative ValuesIn Exercises 67–72, find the value of (f ∘ g)' at the given value of x.f(u) = cot(πu/10), u = g(x) = 5√x, x = 1