In Exercises 41–58, find dy/dt.


y = sin²(πt − 2)

Verified step by step guidance

1. Identify the function y = sin²(πt − 2). This can be rewritten as y = (sin(πt − 2))² to make it easier to differentiate.
2. Recognize that you need to use the chain rule to differentiate y with respect to t. The chain rule states that if you have a composite function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
3. Differentiate the outer function u² with respect to u, where u = sin(πt − 2). The derivative of u² with respect to u is 2u.
4. Differentiate the inner function sin(πt − 2) with respect to t. The derivative of sin(πt − 2) is cos(πt − 2) multiplied by the derivative of the inside function πt − 2, which is π.
5. Combine the results from the previous steps using the chain rule: dy/dt = 2(sin(πt − 2)) * cos(πt − 2) * π.

Key Concepts

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(t)) is composed of two functions, the derivative dy/dt 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 t. This is essential for differentiating y = sin²(πt − 2).

Derivative of Sine Function

Understanding the derivative of the sine function is crucial, as it forms the basis for differentiating trigonometric expressions. The derivative of sin(x) with respect to x is cos(x). In the context of y = sin²(πt − 2), recognizing that the derivative of sin(πt − 2) is cos(πt − 2) is key to applying the chain rule effectively.

Power Rule

The power rule is a basic differentiation rule used when differentiating functions of the form u^n, where u is a function of t. It states that the derivative is n*u^(n-1) * du/dt. For y = sin²(πt − 2), this rule helps in differentiating the squared term, where n is 2, and u is sin(πt − 2), requiring the use of the chain rule for du/dt.