Applications


Suppose that f(x) = d/dx (1 − √x) and g(x) = d/dx (x + 2).


Find:


∫[−f(x)] dx

Verified step by step guidance

1. First, identify the functions f(x) and g(x) as derivatives given in the problem: \( f(x) = \frac{d}{dx} (1 - \sqrt{x}) \) and \( g(x) = \frac{d}{dx} (x + 2) \).
2. Calculate \( f(x) \) by differentiating the function inside: \( 1 - \sqrt{x} = 1 - x^{1/2} \). Use the power rule for differentiation: \( \frac{d}{dx} x^{n} = n x^{n-1} \).
3. Express \( f(x) \) explicitly as \( f(x) = 0 - \frac{1}{2} x^{-1/2} = -\frac{1}{2 \sqrt{x}} \).
4. The integral to find is \( \int -f(x) \, dx \). Substitute \( f(x) \) into the integral to get \( \int -\left(-\frac{1}{2 \sqrt{x}}\right) dx = \int \frac{1}{2 \sqrt{x}} \, dx \).
5. Rewrite the integral in terms of exponents: \( \int \frac{1}{2} x^{-1/2} \, dx \). Use the power rule for integration: \( \int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C \), where \( n \neq -1 \).

Key Concepts

Derivative and Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change or slope at any point. In this problem, f(x) and g(x) are defined as derivatives of given functions, so understanding how to compute derivatives is essential.

Integral and Antiderivative

Integration is the reverse process of differentiation, used to find the original function from its derivative. The integral ∫[−f(x)] dx requires finding the antiderivative of the negative of f(x), which involves applying integration rules to the derived function.

Properties of Square Roots and Power Functions

The function inside the derivative includes a square root, which can be rewritten as a power function (x^(1/2)). Understanding how to differentiate and integrate power functions, including fractional exponents, is crucial for correctly solving the problem.