Find the derivatives of the functions in Exercises 1–42.


𝔂 = (θ² + sec θ + 1)³

Verified step by step guidance

1. Identify the function: We have 𝔂 = (θ² + sec θ + 1)³. This is a composite function, where the outer function is a cube and the inner function is θ² + sec θ + 1.
2. Apply the chain rule: The chain rule states that if you have a composite function y = f(g(x)), then the derivative y' = f'(g(x)) * g'(x). Here, f(u) = u³ and g(θ) = θ² + sec θ + 1.
3. Differentiate the outer function: The derivative of f(u) = u³ with respect to u is f'(u) = 3u². Substitute u = θ² + sec θ + 1 into this derivative.
4. Differentiate the inner function: Find the derivative of g(θ) = θ² + sec θ + 1 with respect to θ. The derivative of θ² is 2θ, and the derivative of sec θ is sec θ tan θ.
5. Combine the derivatives: Multiply the derivative of the outer function by the derivative of the inner function according to the chain rule. This gives you the derivative of the original function 𝔂 with respect to θ.

Key Concepts

Derivative

The derivative of a function measures how the function's output value changes as its input value changes. It is a fundamental concept in calculus that provides the slope of the tangent line to the curve at any given point. The derivative can be computed using various rules, such as the power rule, product rule, and chain rule.

Chain Rule

The chain rule is a formula for computing the derivative of the composition of two or more functions. It states that if you have a function that is composed of another function, the derivative can be found by multiplying the derivative of the outer function by the derivative of the inner function. This is particularly useful when dealing with functions raised to a power, as in the given problem.

Secant Function

The secant function, denoted as sec(θ), is the reciprocal of the cosine function, defined as sec(θ) = 1/cos(θ). Understanding the secant function is essential when differentiating functions that include it, as it requires the application of the quotient rule and knowledge of trigonometric derivatives. Its behavior and properties are crucial for accurately finding derivatives involving trigonometric functions.