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

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Identify the function given: \(y = e^{5 - 7x}\). This is an exponential function where the exponent is a linear expression in \(x\).

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.

Set \(u(x) = 5 - 7x\). Next, find the derivative of \(u(x)\) with respect to \(x\): \(\frac{du}{dx} = -7\).

Apply the chain rule by multiplying the original function by the derivative of the exponent: \(\frac{dy}{dx} = e^{5 - 7x} \cdot (-7)\).

Write the final expression for the derivative as \(\frac{dy}{dx} = -7 e^{5 - 7x}\) (do not simplify further if not required).

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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.

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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.

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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.

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