Find the function f given that the slope of the tangent line...

Find the function $f$ given that the slope of the tangent line to the graph of fat any point $(x, f(x))$ is $f^{\prime}(x)$ and that the graph of $f$ passes through the given point.
$$f^{\prime}(x)=\frac{1}{2} x^{-1 / 2} ;(2, \sqrt{2})$$

Key Concepts

Derivative and Tangent Line

The derivative of a function at a given point represents the slope of the tangent line to the curve at that point. This concept is fundamental in understanding how a function changes and is central to the studies of rates of change and instantaneous behavior in calculus.

Antiderivative (Integration)

The process of integration, or finding the antiderivative, is essentially the reverse of differentiation. It involves determining the original function given its derivative, crucial for solving differential equations and recovering functions from their rates of change.

Power Rule for Integration

The power rule is a basic technique in integration that applies to functions of the form x^n. It states that the integral of x^n is x^(n+1) divided by (n+1), provided n is not -1. This rule simplifies finding antiderivatives for functions that follow a polynomial or power structure.

Constant of Integration

When finding an antiderivative, an arbitrary constant is added because derivatives of constant terms are zero. This addition is known as the constant of integration and accounts for the family of solutions that differ by a constant value.

Initial Condition (Boundary Value)

The initial condition is an extra piece of information that specifies the value of the unknown function at a particular point. It is used to determine the specific solution from the family of antiderivatives by solving for the constant of integration.

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