In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
33. y = ln(sec(lnθ))

Verified step by step guidance

1. Identify the outer function and the inner functions in the composition. Here, the outer function is the natural logarithm \(y = \ln(u)\) where \(u = \sec(v)\), and the inner function is \(v = \ln\theta\).
2. Apply the chain rule for derivatives: \(\frac{dy}{d\theta} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{d\theta}\).
3. Compute each derivative separately: \(\frac{dy}{du} = \frac{1}{u}\), \(\frac{du}{dv} = \sec(v) \tan(v)\) (since the derivative of \(\sec x\) is \(\sec x \tan x\)), and \(\frac{dv}{d\theta} = \frac{1}{\theta}\) (derivative of \(\ln \theta\)).
4. Substitute back the expressions for \(u\) and \(v\) to write the derivative in terms of \(\theta\): \(\frac{dy}{d\theta} = \frac{1}{\sec(\ln \theta)} \cdot \sec(\ln \theta) \tan(\ln \theta) \cdot \frac{1}{\theta}\).
5. Simplify the expression by canceling terms where possible to get the final derivative formula in terms of \(\theta\).

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 is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).

Derivative of Logarithmic Functions

The derivative of the natural logarithm function ln(u) with respect to its variable is 1/u times the derivative of u. Specifically, if y = ln(u), then dy/dx = (1/u) * du/dx. This rule is essential when differentiating functions involving logarithms.

Derivative of Trigonometric Functions

Knowing the derivatives of trigonometric functions like secant is crucial. The derivative of sec(x) is sec(x)tan(x). When the argument is a function of the variable, the chain rule applies, multiplying by the derivative of the inner function.