Higher-order derivatives Find f′(x),f′′(x), and f′′′(x).
f(x) = 1/x

검증된 단계별 안내

Step 1: Identify the function f(x) = \(\frac{1}{x}\). This can be rewritten using exponent notation as f(x) = x^{-1}.

Step 2: Find the first derivative f'(x) by applying the power rule. The power rule states that if f(x) = x^n, then f'(x) = n \(\cdot\) x^{n-1}. For f(x) = x^{-1}, f'(x) = -1 \(\cdot\) x^{-2}.

Step 3: Simplify the expression for the first derivative. f'(x) = -x^{-2} can be rewritten as f'(x) = -\(\frac{1}{x^2}\).

Step 4: Find the second derivative f''(x) by differentiating f'(x) = -x^{-2} again using the power rule. f''(x) = 2 \(\cdot\) x^{-3}.

Step 5: Simplify the expression for the second derivative. f''(x) = 2x^{-3} can be rewritten as f''(x) = \(\frac{2}{x^3}\).

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주요 개념

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

First Derivative

The first derivative of a function, denoted as f'(x), represents the rate of change of the function with respect to its variable. For the function f(x) = 1/x, the first derivative can be found using the power rule or quotient rule, indicating how the function's value changes as x varies.

Second Derivative

The second derivative, denoted as f''(x), is the derivative of the first derivative. It provides information about the curvature of the function and can indicate concavity. For f(x) = 1/x, calculating the second derivative involves differentiating f'(x) again, revealing how the rate of change itself is changing.