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

Guida verificata passo dopo passo

Identify the function given: \(y = x^{\pi}\), where \(\pi\) is a constant approximately equal to 3.14159.

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

Apply the power rule to the function: since \(n = \pi\), the derivative is \(\frac{dy}{dx} = \pi x^{\pi - 1}\).

Write the final expression for the derivative without simplifying the exponent further: \(\frac{dy}{dx} = \pi x^{\pi - 1}\).

Note that this derivative is valid for \(x > 0\) because the base \(x\) is raised to an irrational power \(\pi\).

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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.

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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.

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