In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)

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Identify the function given: \(y = 3^{\log_{2} t}\). Here, the base of the exponent is 3, and the exponent is \(\log_{2} t\).

Recall that when differentiating an exponential function with a variable exponent, it is often helpful to rewrite the function using the natural exponential and logarithm: \(y = e^{\ln(3^{\log_{2} t})}\).

Use the logarithm power rule to simplify the exponent inside the natural exponential: \(\ln(3^{\log_{2} t}) = \log_{2} t \cdot \ln(3)\), so \(y = e^{\log_{2} t \cdot \ln(3)}\).

Differentiate \(y\) with respect to \(t\) using the chain rule: \(\frac{dy}{dt} = y \cdot \frac{d}{dt} (\log_{2} t \cdot \ln(3))\). Since \(\ln(3)\) is constant, focus on differentiating \(\log_{2} t\).

Recall that \(\log_{2} t = \frac{\ln t}{\ln 2}\), so \(\frac{d}{dt} \log_{2} t = \frac{1}{t \ln 2}\). Substitute this back to get \(\frac{dy}{dt} = 3^{\log_{2} t} \cdot \ln(3) \cdot \frac{1}{t \ln 2}\).

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Key Concepts

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

Logarithmic Functions and Change of Base

Logarithmic functions are the inverses of exponential functions. The change of base formula allows rewriting logarithms with any base into a more convenient base, such as natural logarithms, which simplifies differentiation.

Exponential Functions with Variable Exponents

When the exponent of an exponential function is itself a function of the variable, differentiation requires applying the chain rule and often rewriting the expression using natural exponentials and logarithms.

Chain Rule in Differentiation

The chain rule is used to differentiate 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.

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