In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
9. y = 8^(-t)

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Identify the function given: \(y = 8^{-t}\). Here, the base is a constant (8) and the exponent is a function of \(t\) (specifically, \(-t\)).

Recall the derivative rule for exponential functions with a constant base: If \(y = a^{u}\) where \(a\) is a positive constant and \(u\) is a function of \(t\), then the derivative is \(\frac{dy}{dt} = a^{u} \cdot \ln(a) \cdot \frac{du}{dt}\).

In this problem, set \(a = 8\) and \(u = -t\). Next, find the derivative of the exponent \(u\) with respect to \(t\): \(\frac{du}{dt} = \frac{d}{dt}(-t) = -1\).

Apply the formula: \(\frac{dy}{dt} = 8^{-t} \cdot \ln(8) \cdot (-1)\).

Simplify the expression if needed, keeping the derivative in terms of \(t\) and constants, but do not calculate the numerical value of \(\ln(8)\).

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Key Concepts

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

Derivative of Exponential Functions

The derivative of an exponential function a^x, where a is a positive constant, is found using the formula d/dx(a^x) = a^x * ln(a). This rule applies when the exponent is a variable, allowing us to differentiate expressions like 8^(-t).

Chain Rule

The chain rule is used to differentiate composite functions. When the exponent is a function of the variable, such as -t, the derivative is the derivative of the outer function times the derivative of the inner function. For y = 8^(-t), we multiply by the derivative of -t.

Natural Logarithm (ln)

The natural logarithm ln(a) appears in the derivative of exponential functions with base a. It is the logarithm to the base e and is essential for expressing the rate of change of functions like 8^(-t), since the derivative involves multiplying by ln(8).

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