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.
$$f^{\prime}(x)=\frac{2}{x}+1 ;(1,2)$$

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Key Concepts

Initial Value Problem

An initial value problem involves solving a differential equation together with a specific condition that the solution must satisfy at a given point. This additional condition is essential for determining the constant of integration and ensuring that the solution is uniquely specified.

Antiderivative

An antiderivative is a function that, when differentiated, yields the original function. The process of finding an antiderivative is known as integration, and it is fundamental to recovering a function from its rate of change.

Constant of Integration

When taking an indefinite integral, an arbitrary constant is added because the derivative of any constant is zero. This constant can be determined using additional information, such as an initial condition or boundary value, to obtain a unique solution.

Derivative as Slope

The derivative of a function at a given point represents the slope of the tangent line to the graph at that point. This concept connects the geometric interpretation of the derivative with its use in understanding how a function changes.

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