Find the derivatives of the functions in Exercises 19–40.

q = sin(t / (√t + 1))

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Identify the function: q = sin(t / (√t + 1)). This is a composition of functions, so we will need to use the chain rule and the quotient rule.

Apply the chain rule: The outer function is sin(u), where u = t / (√t + 1). The derivative of sin(u) with respect to u is cos(u).

Apply the quotient rule to find the derivative of u = t / (√t + 1): The quotient rule states that if you have a function v/w, its derivative is (v'w - vw') / w². Here, v = t and w = √t + 1.

Find the derivatives of v and w: v' = 1 and w' = (1/2)t^(-1/2) (using the power rule for √t).

Substitute the derivatives into the quotient rule: The derivative of u = (1(√t + 1) - t(1/2)t^(-1/2)) / (√t + 1)². Simplify this expression and then multiply by cos(u) to find the derivative of q with respect to t.

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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. It states that if a function y = f(g(x)) is composed of two functions, the derivative is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to x. This rule is essential when dealing with nested functions like sin(t / (√t + 1)).

Quotient Rule

The quotient rule is used to differentiate functions that are ratios of two differentiable functions. If you have a function h(x) = u(x)/v(x), the derivative h'(x) is given by (v(x)u'(x) - u(x)v'(x)) / (v(x))^2. This rule is crucial for differentiating the function t / (√t + 1) in the given problem.

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions is vital for solving calculus problems involving these functions. The derivative of sin(x) is cos(x), which is a key fact needed when differentiating the function q = sin(t / (√t + 1)). This knowledge allows us to apply the chain rule effectively to find the derivative of the composite trigonometric function.

Find the derivatives of the functions in Exercises 19–40.q = sin(t / (√t + 1))