In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
11. y = 5x^(3.6)

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Identify the function given: \(y = 5x^{3.6}\), where \(y\) is expressed in terms of \(x\).

Recall the power rule for differentiation: if \(y = ax^n\), then \(\frac{dy}{dx} = a n x^{n-1}\).

Apply the power rule to the function: multiply the coefficient 5 by the exponent 3.6 to get the new coefficient.

Subtract 1 from the exponent 3.6 to find the new exponent for \(x\) in the derivative.

Write the derivative as \(\frac{dy}{dx} = 5 \times 3.6 \times x^{3.6 - 1}\).

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주요 개념

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Power Rule for Differentiation

The power rule states that the derivative of x raised to a constant power n is n times x to the (n-1) power. It is expressed as d/dx[x^n] = n*x^(n-1), and applies to any real number exponent, including decimals.

Constant Multiple Rule

When differentiating a function multiplied by a constant, the constant can be factored out and multiplied by the derivative of the function. For example, d/dx[c*f(x)] = c * d/dx[f(x)], where c is a constant.

Derivative Notation and Variables

Understanding the notation dy/dx means the derivative of y with respect to x, indicating the rate of change of y as x changes. Correctly identifying the variable of differentiation is essential for applying derivative rules properly.

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.11. y = 5x^(3.6)