"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
67. y = 7^(sec θ) ln 7"

Verified step by step guidance

1. Identify the function to differentiate: \(y = 7^{\sec \theta} \ln 7\). Notice that \(\ln 7\) is a constant multiplier.
2. Rewrite the function to clarify the structure: \(y = (7^{\sec \theta}) \cdot (\ln 7)\), where \(\ln 7\) is constant with respect to \(\theta\).
3. Recall the derivative formula for an exponential function with a variable exponent: If \(y = a^{u(\theta)}\), then \(\frac{dy}{d\theta} = a^{u(\theta)} \ln a \cdot \frac{du}{d\theta}\).
4. Apply the formula with \(a = 7\) and \(u(\theta) = \sec \theta\). Compute \(\frac{du}{d\theta} = \frac{d}{d\theta} (\sec \theta) = \sec \theta \tan \theta\).
5. Combine all parts to write the derivative: \(\frac{dy}{d\theta} = \ln 7 \cdot 7^{\sec \theta} \cdot \ln 7 \cdot \sec \theta \tan \theta\). Simplify by multiplying constants where appropriate.

Key Concepts

Derivative of Exponential Functions with Variable Exponents

When differentiating functions where the exponent is a variable expression, such as a^g(x), we use the formula d/dx[a^g(x)] = a^g(x) * ln(a) * g'(x). This combines the chain rule with the natural logarithm of the base to handle the variable exponent.

Chain Rule

The chain rule is used to differentiate composite functions. It states that the derivative of f(g(x)) is f'(g(x)) * g'(x). In this problem, it applies to differentiating sec(θ) inside the exponent.

Derivative of Trigonometric Functions

Knowing the derivatives of trigonometric functions is essential. Specifically, the derivative of sec(θ) with respect to θ is sec(θ) tan(θ). This is needed to find g'(θ) when differentiating the exponent.