Second Derivatives

Find y'' in Exercises 59–64.

y = (1 − √x)⁻¹

검증된 단계별 안내

First, rewrite the given function \( y = (1 - \sqrt{x})^{-1} \) in a more convenient form for differentiation. This can be expressed as \( y = \frac{1}{1 - \sqrt{x}} \).

To find the first derivative \( y' \), apply the chain rule and the power rule. Start by differentiating the outer function \( u^{-1} \) where \( u = 1 - \sqrt{x} \). The derivative of \( u^{-1} \) is \( -u^{-2} \).

Next, differentiate the inner function \( u = 1 - \sqrt{x} \). The derivative of \( \sqrt{x} \) is \( \frac{1}{2\sqrt{x}} \), so the derivative of \( u \) is \( -\frac{1}{2\sqrt{x}} \).

Combine these results using the chain rule: \( y' = -\frac{1}{(1 - \sqrt{x})^2} \cdot (-\frac{1}{2\sqrt{x}}) \). Simplify this expression to find \( y' \).

To find the second derivative \( y'' \), differentiate \( y' \) with respect to \( x \) again. Use the quotient rule or product rule as needed, and simplify the resulting expression to obtain \( y'' \).

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주요 개념

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

Chain Rule

The chain rule is a fundamental technique in calculus used to differentiate composite functions. It states that if a function y = f(g(x)) is composed of two functions, the derivative y' is found by multiplying the derivative of the outer function f with respect to the inner function g by the derivative of the inner function g with respect to x. This rule is essential when dealing with nested functions, such as y = (1 − √x)⁻¹.

Power Rule

The power rule is a basic rule for differentiation that applies to functions of the form y = x^n, where n is any real number. The derivative of y with respect to x is given by y' = n*x^(n-1). This rule is crucial for differentiating terms like (1 − √x)⁻¹, where the exponent is negative, and it helps simplify the process of finding the first and second derivatives.