### Video Transcript

Find the linear approximation of the function π of π₯ equals the square root of π₯ at π₯ equals four.

Linear approximation, sometimes called the tangent line approximation, is a way of approximating values of π of π₯ as long as we stay fairly near a given number π₯ equals π. The formula we use is π of π₯ equals π of π plus π prime of π times π₯ minus π. In our question, π of π₯ is equal to the square root of π₯. So weβll begin by working out what π prime of π₯ is, so we can use that to evaluate π prime of π.

Now remember, π prime of π₯ is the first derivative of the function with respect to π₯. Itβs sensible to write the square root of π₯ as π₯ to the power of a half before finding its derivative. Then we use the general result that the derivative of π₯ to the power of π for real nonzero constants π is π times π₯ to the power of π minus one. And we see that π prime of π₯ is a half times π₯ to the power of a half minus one. Thatβs a half times π₯ to the power of negative one-half.

Weβre trying to work out the linear approximation of our function at the point where π₯ is equal to four. This is the value of π in our equation, so we let π be equal to four. Then we see that π of π is π of four and π prime of π is π prime of four. Of course, π of four is just the square root of four, which is two. And π prime of four is a half times four to the power of negative one-half, which is equivalent to saying a half times one over the square root of four. Now, the square of four is two, so a half times one over the square root of four is a quarter.

Letβs substitute everything we have into our formula for the tangent line approximation. π of π is two and π prime of π is a quarter. Then π₯ minus π is π₯ minus four. We distribute our parentheses. And we get two plus a quarter π₯ minus a quarter times four which is two plus a quarter π₯ minus one. Simplifying and we obtain the tangent line approximation to be equal to a quarter π₯ plus one.