Evaluate and simplify y'.
y = 2x^√2

검증된 단계별 안내

Step 1: Recognize that the function y = 2x^{\(\sqrt{2}\)} is in the form of a power function, where the exponent is an irrational number, \(\sqrt{2}\).

Step 2: To differentiate y with respect to x, use the power rule for differentiation, which states that if y = x^n, then y' = n \(\cdot\) x^{n-1}.

Step 3: Apply the power rule to the given function. Here, n = \(\sqrt{2}\), so the derivative y' = \(\sqrt{2}\) \(\cdot\) 2x^{\(\sqrt{2}\) - 1}.

Step 4: Simplify the expression for y' by multiplying the constant \(\sqrt{2}\) with 2, resulting in y' = 2\(\sqrt{2}\) \(\cdot\) x^{\(\sqrt{2}\) - 1}.

Step 5: The expression y' = 2\(\sqrt{2}\) \(\cdot\) x^{\(\sqrt{2}\) - 1} is the simplified form of the derivative of the given function.

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주요 개념

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

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. The derivative represents the rate of change of the function with respect to its variable. In this case, we need to differentiate the function y = 2x^(√2) to find y'.

Power Rule

The Power Rule is a basic rule for differentiation that states if y = x^n, then the derivative y' = n*x^(n-1). This rule simplifies the process of finding derivatives for polynomial and power functions, making it essential for evaluating y' in the given function.

Constant Multiple Rule

The Constant Multiple Rule states that if a function is multiplied by a constant, the derivative of the function is the constant multiplied by the derivative of the function itself. In this case, since y = 2x^(√2), we will apply this rule to factor out the constant 2 when differentiating.

Evaluate and simplify y'.y = 2x√2