In Exercises 59–86, find the derivative of y with respect to the given independent variable.
71. y = log₂(5θ)

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Recall the formula for the derivative of a logarithm with an arbitrary base: if \( y = \log_a(u) \), then \( \frac{dy}{dx} = \frac{1}{u \ln(a)} \cdot \frac{du}{dx} \), where \( \ln \) is the natural logarithm.

Identify the base \( a \) and the argument \( u \) in the given function. Here, \( y = \log_2(5\theta) \), so \( a = 2 \) and \( u = 5\theta \).

Compute the derivative of the inside function \( u = 5\theta \) with respect to \( \theta \). Since \( 5 \) is a constant, \( \frac{du}{d\theta} = 5 \).

Apply the derivative formula: \( \frac{dy}{d\theta} = \frac{1}{5\theta \ln(2)} \times 5 \).

Simplify the expression by canceling common factors if possible to express the derivative in its simplest form.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Derivative of Logarithmic Functions

The derivative of a logarithmic function depends on its base. For a logarithm with base a, log_a(x), the derivative is 1/(x ln(a)) times the derivative of x. Understanding this allows you to differentiate log₂(5θ) by applying the chain rule.

Chain Rule

The chain rule is used to differentiate composite functions. When a function is inside another, like 5θ inside the logarithm, you first differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function.

Properties of Logarithms

Logarithmic properties, such as log_a(bx) = log_a(b) + log_a(x), can simplify differentiation. Recognizing these properties helps break down complex logarithmic expressions into simpler parts, making differentiation more straightforward.

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