In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(-5x)

Verified step by step guidance

1. Identify the function given: \(y = e^{-5x}\), where \(y\) is expressed in terms of \(x\).
2. Recall the chain rule for differentiation: if \(y = e^{u(x)}\), then \(\frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx}\).
3. Set the inner function \(u(x) = -5x\) and find its derivative with respect to \(x\): \(\frac{du}{dx} = -5\).
4. Apply the chain rule by multiplying the derivative of the outer function by the derivative of the inner function: \(\frac{dy}{dx} = e^{-5x} \cdot (-5)\).
5. Write the final expression for the derivative as \(\frac{dy}{dx} = -5 e^{-5x}\) (do not calculate the numerical value).

Key Concepts

Derivative of Exponential Functions

The derivative of an exponential function with base e, such as e^u, is found by multiplying e^u by the derivative of the exponent u. This rule is essential for differentiating functions like y = e^(-5x), where the exponent is a function of x.

Chain Rule

The chain rule is used to differentiate composite functions. When a function is composed of an outer function and an inner function, the derivative is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. For y = e^(-5x), the inner function is -5x.

Notation and Variable Dependence

Understanding which variable to differentiate with respect to (x, t, or θ) is crucial. In this problem, y is expressed in terms of x, so differentiation is with respect to x. Recognizing the independent variable ensures correct application of differentiation rules.