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

Verified step by step guidance

1. Identify the function given: \(y = e^{5 - 7x}\). This is an exponential function where the exponent is a linear expression in \(x\).
2. Recall the chain rule for differentiation: if \(y = e^{u(x)}\), then \(\frac{dy}{dx} = e^{u(x)} \cdot \frac{du}{dx}\), where \(u(x)\) is the exponent function.
3. Set \(u(x) = 5 - 7x\). Next, find the derivative of \(u(x)\) with respect to \(x\): \(\frac{du}{dx} = -7\).
4. Apply the chain rule by multiplying the original function by the derivative of the exponent: \(\frac{dy}{dx} = e^{5 - 7x} \cdot (-7)\).
5. Write the final expression for the derivative as \(\frac{dy}{dx} = -7 e^{5 - 7x}\) (do not simplify further if not required).

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 uses the chain rule and reflects how the rate of change depends on both the function and its exponent.

Chain Rule

The chain rule is a method for differentiating composite functions. It states that the derivative of f(g(x)) is f'(g(x)) times g'(x). This is essential when the exponent itself is a function of x, like 5 - 7x in this problem.

Basic Differentiation Rules

Understanding how to differentiate constants and linear functions is fundamental. For example, the derivative of a constant is zero, and the derivative of -7x is -7. These rules help simplify the derivative of the exponent in the given function.