If r = sin(f(t)), f(0) = π/3, and f'(0) = 4, then what is dr/dt at t = 0?

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First, identify the given functions and their derivatives. We have r = sin(f(t)), and we need to find dr/dt at t = 0.

Use the chain rule to differentiate r with respect to t. The chain rule states that if r = sin(f(t)), then dr/dt = cos(f(t)) * f'(t).

Substitute the given values into the derivative. We know f(0) = π/3 and f'(0) = 4.

Evaluate cos(f(t)) at t = 0. Since f(0) = π/3, we have cos(f(0)) = cos(π/3).

Finally, calculate dr/dt at t = 0 by substituting cos(π/3) and f'(0) = 4 into the expression dr/dt = cos(f(t)) * f'(t).

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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 concept in calculus used to differentiate composite functions. If you have a function y = g(f(x)), the derivative dy/dx is found by multiplying the derivative of the outer function g with respect to the inner function f, g'(f(x)), by the derivative of the inner function f with respect to x, f'(x). This rule is essential for finding dr/dt when r is a function of f(t).

Implicit Differentiation

Implicit differentiation is a technique used when a function is not explicitly solved for one variable in terms of another. In this problem, r is given as a function of f(t), and we need to differentiate r with respect to t. By applying implicit differentiation, we can find dr/dt by differentiating both sides of the equation r = sin(f(t)) with respect to t, using the chain rule.

Trigonometric Derivatives

Understanding the derivatives of trigonometric functions is crucial for solving this problem. The derivative of sin(x) with respect to x is cos(x). In the context of this problem, when differentiating r = sin(f(t)) with respect to t, we use this derivative to find dr/dt, which involves multiplying cos(f(t)) by the derivative of the inner function f(t) with respect to t.