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

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Identify the function to differentiate: \(y = 7^{\sec \theta} \ln 7\). Notice that \(\ln 7\) is a constant multiplier.

Rewrite the function to clarify the structure: \(y = (7^{\sec \theta}) \cdot (\ln 7)\), where \(\ln 7\) is constant with respect to \(\theta\).

Recall the derivative formula for an exponential function with a variable exponent: If \(y = a^{u(\theta)}\), then \(\frac{dy}{d\theta} = a^{u(\theta)} \ln a \cdot \frac{du}{d\theta}\).

Apply the formula with \(a = 7\) and \(u(\theta) = \sec \theta\). Compute \(\frac{du}{d\theta} = \frac{d}{d\theta} (\sec \theta) = \sec \theta \tan \theta\).

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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