In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
7. y = log₂(x²/2)

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Recall that the function is given as \(y = \log_2\left(\frac{x^2}{2}\right)\). Our goal is to find \(\frac{dy}{dx}\), the derivative of \(y\) with respect to \(x\).

Use the change of base formula for logarithms to rewrite the function in terms of natural logarithms: \(y = \frac{\ln\left(\frac{x^2}{2}\right)}{\ln(2)}\).

Since \(\ln(2)\) is a constant, the derivative of \(y\) with respect to \(x\) is \(\frac{1}{\ln(2)}\) times the derivative of \(\ln\left(\frac{x^2}{2}\right)\) with respect to \(x\).

Apply the chain rule to differentiate \(\ln\left(\frac{x^2}{2}\right)\). The derivative of \(\ln(u)\) with respect to \(x\) is \(\frac{1}{u} \cdot \frac{du}{dx}\), where \(u = \frac{x^2}{2}\).

Calculate \(\frac{du}{dx}\) where \(u = \frac{x^2}{2}\). Then, combine all parts to express \(\frac{dy}{dx} = \frac{1}{\ln(2)} \cdot \frac{1}{\frac{x^2}{2}} \cdot \frac{du}{dx}\).

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Key Concepts

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

Logarithmic Functions and Their Properties

Logarithmic functions are the inverses of exponential functions. Understanding properties like the change of base formula and the rules for logarithms (product, quotient, and power) helps simplify expressions before differentiation.

Derivative of Logarithmic Functions

The derivative of a logarithmic function depends on its base. For log base a, the derivative of log_a(u) with respect to x is (1 / (u ln a)) * du/dx, where u is a function of x. This formula is essential for differentiating log₂(x²/2).

Chain Rule

The chain rule is used to differentiate composite functions. When differentiating log₂(x²/2), you treat the inside function (x²/2) separately and multiply its derivative by the derivative of the outer logarithmic function.

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