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Find an equation of the tangent line to the curve at the given point.

$ y = \sqrt{x} $, $ (1, 1) $

$y=\frac{1}{2} x+\frac{1}{2}$

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Campbell University

Oregon State University

Idaho State University

Okay, we have the function Y equals the square root of X. And we want to find the equation of the line that is tangent to the curve to the graph of this function at the 0.1 comma one. Uh So we are going to use what's called the point slope form of the equation of a line to find the equation of the tangent line. Uh First since it's gonna be tension to the graph at the 0.1 comma one. Uh We're gonna label these X and Y coordinates as X one and Y one. And the point slope form. The equation of a line learned back in algebra Is why- Why one equals M the slope of our tangent line. We'll talk about that in a minute Uh Times X -X one. So the Y in the X are going to stay Y and X the X one and Y one. You already know their values from the coordinates of the point. All you have to do now is find m is the slope of the tangent line. Well, the slope of the tangent line is going to be the derivative of our function at this value of X when x is one. So we have to find why crime. Well, the first thing I'm gonna do is I'm going to rewrite our Y function instead of writing as square defects. I'm going to write it as X to the one half power, which is equivalent to the square root effects. And so why prime using uh the rules for derivatives, the power rule. Uh the derivative of extra one half is going to be one half times X. And then we have to subtract one from this experience of one half to attract one is negative one half. And so if we wanted to rewrite this a little bit neater, uh we can write this as one over, we have to in the denominator and then extend the negative one half is really extra deposit of one half. If we move it down to the denominator, an extra deposit of one half can be just rewritten as square root of X. So that is why prime. Now we said that M is going to be the derivative of our function evaluated at this X value when X is one. So why prime evaluated at one Is going to be one over Instead of two times the square root of X. two times the square root of one square root of one is just one, one times two is just too. So this is 1/2. So that's our derivative of the function when x is one, that is going to be our M. So the X one is going to go in here. Uh why one is going to go in here And the derivative of our functions when X is one came out to be one half, that's going to be the slope of our tangent line. So the equation of our tangent line, why subtract, why one, why -1 Equals and which is 1/2 at times? Princess X minus X. One X. subtract X one which is one. We're going to distribute the multiplication. Just do a little bit algebra now. So why should attract one equals one half times X minus one is one half X. Subtract one half times one is one half last but not least. We're gonna get why by itself. Um By adding one to both sides of the equation. So we get why equals one half X. And then when we add one to both sides minus one half plus one is plus one half. So final answer here is the equation of our change in line. So this is the equation of a line that would be tangent to the curve to the graph of this function at this point.