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

검증된 단계별 안내

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 \(\pi\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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