In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
1. y = 10e^(-x/5)

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Identify the function given: \(y = 10e^{-\frac{x}{5}}\).

Recall the derivative rule for exponential functions: if \(y = e^{u(x)}\), then \(\frac{dy}{dx} = e^{u(x)} \cdot u'(x)\).

Set the inner function \(u(x) = -\frac{x}{5}\) and find its derivative: \(u'(x) = -\frac{1}{5}\).

Apply the chain rule: \(\frac{dy}{dx} = 10 \cdot e^{-\frac{x}{5}} \cdot \left(-\frac{1}{5}\right)\).

Simplify the expression by multiplying constants to express the derivative clearly.

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주요 개념

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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 exponential growth or decay rates change with respect to the variable.

Chain Rule

The chain rule is a method for differentiating 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.

Constant Multiple Rule

The constant multiple rule states that the derivative of a constant multiplied by a function is the constant multiplied by the derivative of the function. This simplifies differentiation when constants are involved.

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.1. y = 10e^(-x/5)