In Exercises 41–58, find dy/dt.


y = (1 + cos(2t))⁻⁴

Verified step by step guidance

1. Identify the function y = (1 + cos(2t))^(-4) and recognize that you need to find the derivative dy/dt.
2. Apply the chain rule for differentiation. The chain rule states that if you have a composite function y = f(g(t)), then dy/dt = f'(g(t)) * g'(t).
3. Differentiate the outer function f(u) = u^(-4) with respect to u, which gives f'(u) = -4u^(-5).
4. Substitute u = 1 + cos(2t) into the derivative of the outer function, resulting in -4(1 + cos(2t))^(-5).
5. Differentiate the inner function g(t) = 1 + cos(2t) with respect to t. The derivative of cos(2t) is -2sin(2t), so g'(t) = -2sin(2t). Multiply this result by the derivative of the outer function to get dy/dt = -4(1 + cos(2t))^(-5) * (-2sin(2t)).

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, then its derivative dy/dt is f'(g(t)) * g'(t). This rule is essential for finding the derivative of y = (1 + cos(2t))⁻⁴, as it involves differentiating the outer function and the inner function separately.

Trigonometric Derivatives

Understanding the derivatives of trigonometric functions is crucial for solving this problem. The derivative of cos(t) is -sin(t), and when dealing with cos(2t), the chain rule must be applied, resulting in the derivative -2sin(2t). This knowledge is necessary to differentiate the inner function of the composite function y = (1 + cos(2t))⁻⁴.

Power Rule

The power rule is used to differentiate functions of the form y = x^n, where the derivative is n*x^(n-1). In this problem, the function y = (1 + cos(2t))⁻⁴ can be differentiated using the power rule, treating the entire expression inside the parentheses as a single variable. This rule helps in finding the derivative of the outer function in the composite expression.