In Exercises 41–58, find dy/dt.

y = (1 + tan⁴(t/12))³

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Identify the function y = (1 + tan⁴(t/12))³ and recognize that you need to find the derivative dy/dt using the chain rule.

Apply the chain rule: If y = u³, then dy/dt = 3u² * du/dt. Here, u = 1 + tan⁴(t/12).

Find du/dt by differentiating u = 1 + tan⁴(t/12) with respect to t. Use the chain rule again: If u = 1 + v⁴, where v = tan(t/12), then du/dt = 4v³ * dv/dt.

Differentiate v = tan(t/12) with respect to t. Use the chain rule: If v = tan(w), where w = t/12, then dv/dt = sec²(w) * dw/dt. Since w = t/12, dw/dt = 1/12.

Combine all the derivatives: Substitute dv/dt back into du/dt, and then substitute du/dt into dy/dt to get the final expression for dy/dt.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. In this problem, the chain rule helps differentiate y = (1 + tan⁴(t/12))³ by breaking it into manageable parts.

Derivative of Trigonometric Functions

Understanding how to differentiate trigonometric functions is crucial for solving calculus problems involving these functions. The derivative of tan(x) is sec²(x), and this knowledge is essential when differentiating tan⁴(t/12) in the given function. Applying this derivative correctly is key to finding dy/dt.

Power Rule

The power rule is a basic differentiation rule used to find the derivative of functions in the form of xⁿ. It states that the derivative of xⁿ is n*xⁿ⁻¹. In this problem, the power rule is applied to differentiate the expression (1 + tan⁴(t/12))³, which involves raising a function to a power, thus requiring the use of the power rule in conjunction with the chain rule.

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