"In Exercises 59–86, find the derivative of y with respect to the given independent variable.
63. y = x^π"

Verified step by step guidance

1. Identify the function given: \(y = x^{\pi}\), where \(\pi\) is a constant approximately equal to 3.14159.
2. Recall the power rule for differentiation when the exponent is a constant: if \(y = x^n\), then \(\frac{dy}{dx} = n x^{n-1}\).
3. Apply the power rule to the function: since \(n = \pi\), the derivative is \(\frac{dy}{dx} = \pi x^{\pi - 1}\).
4. Write the final expression for the derivative without simplifying the exponent further: \(\frac{dy}{dx} = \pi x^{\pi - 1}\).
5. Note that this derivative is valid for \(x > 0\) because the base \(x\) is raised to an irrational power \(\pi\).

Key Concepts

Derivative of Power Functions

The derivative of a power function y = x^n, where n is a constant, is found using the power rule: dy/dx = n * x^(n-1). This rule applies when the exponent is any real number, including irrational constants like π.

Constant Exponents and Irrational Numbers

When the exponent is a constant, even if irrational like π, it is treated as a fixed number during differentiation. This means the power rule applies directly without additional steps, simplifying the process.

Notation and Differentiation with Respect to the Independent Variable

Differentiating y with respect to x means finding dy/dx, the rate at which y changes as x changes. Understanding this notation is essential for applying differentiation rules correctly to functions of x.