In Exercise, find an equation of the tangent line to the graph of y = f (x) at the given x...

In Exercise, find an equation of the tangent line to the graph of y = f (x) at the given x. Do not apply formula (6), but proceed as we did in Example.

f(x) =

Example

Finding the Equation of the Tangent Line at a Given x Find the slope–point equation of the tangent line to the graph of f(x) =  at x = 2.

SOLUTION In this problem, we are not given a point on the tangent line, but only its first coordinate x = 2. Since the point is on the graph of f(x) =  we get the second coordinate by plugging the x-value into f(x):

Thus,  is the point on the graph and the tangent line. Next, we find the slope of the tangent line. For this purpose, we compute f_(x) by using the power rule:

The slope of the tangent line when x = 2 is

In slope–point form, the equation of the tangent line is y

In general, to find the point–slope equation of the tangent line to the graph of y = f(x) at the point with first coordinate x = a, proceed as follows:

Step 1 Find the point of contact of the graph and the tangent line by evaluating f(x) at x = a. This yields the point (a, f(a)).

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Step 2 Find the slope of the tangent line by evaluating the derivative f′(x) at x = a. This yields the slope m = f′(a).

Using the point (a, f(a)) and the slope m = f′(a), we obtain the equation of the tangent line:

You do not need to memorize formula (6), but you should be able to derive it in a given situation, as we did in Example.