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

Verified step by step guidance

1. Identify the function given: \(y = 8^{-t}\). Here, the base is a constant (8) and the exponent is a function of \(t\) (specifically, \(-t\)).
2. 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 \(\frac{dy}{dt} = a^{u} \cdot \ln(a) \cdot \frac{du}{dt}\).
3. In this problem, set \(a = 8\) and \(u = -t\). Next, find the derivative of the exponent \(u\) with respect to \(t\): \(\frac{du}{dt} = \frac{d}{dt}(-t) = -1\).
4. Apply the formula: \(\frac{dy}{dt} = 8^{-t} \cdot \ln(8) \cdot (-1)\).
5. Simplify the expression if needed, keeping the derivative in terms of \(t\) and constants, but do not calculate the numerical value of \(\ln(8)\).

Key Concepts

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).