In Exercises 41–58, find dy/dt.


y = tan²(sin³(t))

Verified step by step guidance

1. Identify the function y = tan²(sin³(t)). This is a composition of functions, so we'll need to use the chain rule multiple times.
2. Start by differentiating the outermost function with respect to its inner function. The outer function is tan²(u), where u = sin³(t). The derivative of tan²(u) with respect to u is 2tan(u)sec²(u).
3. Next, differentiate the middle function, which is sin³(t) with respect to t. The derivative of sin³(t) with respect to t is 3sin²(t)cos(t), using the chain rule.
4. Now, apply the chain rule: dy/dt = (d/dt)[tan²(sin³(t))] = 2tan(sin³(t))sec²(sin³(t)) * 3sin²(t)cos(t).
5. Simplify the expression by combining the derivatives: dy/dt = 6tan(sin³(t))sec²(sin³(t))sin²(t)cos(t).

Key Concepts

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 f(g(x)) is f'(g(x)) * g'(x). In this problem, the chain rule is essential for differentiating y = tan²(sin³(t)), as it involves nested functions.

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions is crucial for solving this problem. The derivative of sin(t) is cos(t), and the derivative of tan(u) is sec²(u). These derivatives are used in conjunction with the chain rule to find dy/dt for the given function y = tan²(sin³(t)).

Power Rule

The power rule is used to differentiate functions of the form x^n, where the derivative is n*x^(n-1). In this problem, the power rule is applied to tan²(u) and sin³(t) as part of the differentiation process. It helps simplify the expression by reducing the powers during differentiation.