Second Derivatives


Find y'' in Exercises 59–64.


y = (1 − √x)⁻¹

Verified step by step guidance

1. First, rewrite the given function \( y = (1 - \sqrt{x})^{-1} \) in a more convenient form for differentiation. This can be expressed as \( y = \frac{1}{1 - \sqrt{x}} \).
2. 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 \( -u^{-2} \).
3. 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 \( -\frac{1}{2\sqrt{x}} \).
4. 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 \( y' \).
5. 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 \( y'' \).

Key Concepts

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.

Second Derivative

The second derivative, denoted as y'' or d²y/dx², represents the derivative of the first derivative y'. It provides information about the concavity and inflection points of the original function y. Calculating the second derivative involves differentiating the first derivative, often requiring the application of rules like the chain rule and power rule multiple times, especially in complex functions like y = (1 − √x)⁻¹.