Find the equation of the tangent line to the graph of the...

Find the equation of the tangent line to the graph of the given function at the point with the indicated $x$ -coordinate. In each case, sketch the curve together with the appropriate tangent line.
$$
f(x)=\sqrt{x} ; x=4
$$

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Key Concepts

Derivative

The derivative of a function measures the rate at which the function's value changes at a certain point. It represents the slope of the tangent line to the graph of the function at that point, and is fundamental in understanding how the function behaves locally.

Evaluating the Derivative at a Point

Once the general derivative of the function is found, evaluating it at a specific point yields the slope of the tangent line at that point. This process involves substituting the x-coordinate of the point into the derivative formula to obtain the instantaneous rate of change.

Tangent Line Equation

The equation of a tangent line to a function at a particular point can be constructed using the point-slope form of a line. This form utilizes the specific point on the function and the slope obtained from the derivative, providing an equation that approximates the function near that point.

Point-Slope Form

The point-slope form is given by the formula y - y? = m(x - x?), where (x?, y?) is a point on the line and m is the slope. This concept is instrumental in translating the derivative information into the equation of the line that is tangent to the curve.

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