How do I solve this. Find the tangent line of f(x)=\sqrt{x} at x = 25. use the tangent line to estimate square-root 25.1. I know I need the derivative = 1/2\sqrt{x} but then what? Please help!

Hi Ski, and welcome to the Matheno forum!

First, I’m going to point you toward something that will be helpful to review: our pages on finding tangent lines and how they can be used

Absolutely you can start by taking the derivative, but just to clarify, when you write 1 / 2 \sqrt{x} , do you mean

or

I’m going to assume that you meant the **first** expression above, but if you actually meant the second expression, then please let me know. Keep in mind that when writing fractions, it’s less ambiguous to write in the form \frac{a}{b}.

In that case, you are going to want to construct the equation of a **line** tangent to the function where x=25 .

The standard form of a linear equation that runs through the point (x_0, y_0) is y-y_0 = m(x-x_0). The thing that is special about **tangent lines** is that the slope, m, of the equation must be equal to the derivative of the function you are trying to approximate, evaluated at the specific point of tangency. ← (that’s the whole reason for finding the derivative here in the first place)

So, first try to fill out each piece of the standard form linear equation above, using what you know about what m, x_0, and y_0 should be in order for that line to touch the curve \sqrt{x} only once at the point (25,5).

Once you have done this, you can use your linear function to approximate the value of \sqrt{25.1} by simply evaluating your linear function for x=25.1.

Check out Question 6 on https://www.matheno.com/learn/math/calculus-1/tangent-lines-problems-and-solutions/ for a more detailed solution to something very similar. Let us know if this wasn’t enough of a “nudge” in the right direction, if you need a more detailed answer, or if you get stuck again!

Adding my welcome to our forum, @ski. Thanks for posting one of the very first questions here, and please let us know what else we can help you with!

wow thank you Shine!