How to find vertical tangent line implicit differentiation

Explanation:

Given #f(x,y)=x^2+xy+y^2-27=0#

Table of Contents Show

Vertical Tangent in Calculus Example
Graphing & Tables
Video transcript
How do you find the vertical tangent line of a derivative?
What is the equation of a vertical tangent line?

#df=f_x dx + f_y dy = 0#

#dy/dx = - f_x/(f_y) = (2x+y)/(2y+x)#

The horizontal tangent lines have #f_x = 0->x = -y/2# and the vertical tangent lines have #f_y = 0->x = -2y#

So for horizontals

#f(-y/2,y) = y^2/4-2y^2+y^2-27=0->y=pm6#

and for verticals

#f(x,-x/2) = x^2-x^2/2+x^2/4 - 27=0->x=pm 6#

You may want to read this article first: What is a Tangent Line?

Watch the video or read on below:

Vertical Tangent Definition and Example

Can’t see the video? Click here.

A tangent of a curve is a line that touches the curve at one point. It has the same slope as the curve at that point. A vertical tangent touches the curve at a point where the gradient (slope) of the curve is infinite and undefined. On a graph, it runs parallel to the y-axis.

General Steps to find the vertical tangent in calculus and the gradient of a curve:

Find the derivative of the function. The derivative (dy/dx) will give you the gradient (slope) of the curve.
Find a value of x that makes dy/dx infinite; you’re looking for an infinite slope, so the vertical tangent of the curve is a vertical line at this value of x.

Vertical Tangent in Calculus Example

Example Problem: Find the vertical tangent of the curve y = √(x – 2).

Step 1: Differentiate y = √(x – 2). You can use your graphing calculator, or perform the differentiation by hand (using the power rule and the chain rule). I differentiated the function with this online calculator (which also shows you the steps!):

Step 2: Look for values of x that would make dy/dx infinite. This is really where strong algebra skills come in handy, although for this example problem all you need to recognize what happens if you put a “2” into the derivative equation:

Division by zero is undefined. This means that the gradient of the curve is infinite (i.e., vertical) when x = 2.

The vertical tangent of the curve is x = 2.
That’s it!

Graphing & Tables

If you aren’t able to immediately see where your function might return zero, you’ve got two options:

Graphing can sometimes help you see where a vertical tangent line might be.

Graph the function—so you can see where the graph might have a vertical tangent. I used this handy HRW calculator to get the above graph of y = √(x – 2). It’s gairly clear that there’s a vertical tangent at x = 2, though you may want to go through the calculus/algebra anyway to prove it.
Make a table of values and test for several values of x.

The second option can be very time consuming; Strong algebra skills (like knowing when an equation might result in division by zero) will help you to avoid having to make a table.

Tips:

Some curves will have more than one vertical tangent. Always make sure you have found all the values of x that make the gradient infinite. You can use graph-plotting software to check by eye for places where the gradient becomes vertical.
For more tips on where functions might return zero, see the “Guess & Check” section of the Domain and Range article.

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Video transcript

- [Instructor] We're told to consider the curve given by the equation. They give this equation. It can be shown that the derivative of y with respect to x is equal to this expression, and you could figure that out with just some implicit differentiation and then solving for the derivative of y with respect to x. We've done that in other videos. Write the equation of the horizontal line that is tangent to the curve and is above the x-axis. Pause this video, and see if you can have a go at it. So let's just make sure we're visualizing this right. So let me just draw a quick and dirty diagram. If that's my y-axis, this is my x-axis. I don't know exactly what that curve looks like, but imagine you have some type of a curve that looks something like this. Well, there would be two tangent lines that are horizontal based on how I've drawn it. One might be right over there, so it might be like there. And then another one might be maybe right over here. And they want the equation of the horizontal line that is tangent to the curve and is above the x-axis. So what do we know? What is true if this tangent line is horizontal? Well, that tells us that, at this point, dy/dx is equal to zero. In fact, that would be true at both of these points. And we know what dy/dx is. We know that the derivative of y with respect to x is equal to negative two times x plus three over four y to the third power for any x and y. And so when will this equal zero? Well, it's going to equal zero when our numerator is equal to zero and our denominator isn't. So when is our numerator going to be zero? When x is equal to negative three. So when x is equal to negative three, the derivative is equal to zero. So what is going to be the corresponding y value when x is equal to negative three? And, if we know that, well, this equation is just going to be y is equal to something. It's going to be that y value. Well, to figure that out, we just take this x equals negative three, substitute it back into our original equation, and then solve for y. So let's do that. So it's going to be negative three squared plus y to the fourth plus six times negative three is equal to seven. This is nine. This is negative 18. And so we're going to get y to the fourth minus nine is equal to seven, or, adding nine to both sides, we get y to the fourth power is equal to 16. And this would tell us that y is going to be equal to plus or minus two. Well, there would be then two horizontal lines. One would be y is equal to two. The other is y is equal to negative two. But they want us, the equation of the horizontal line that is tangent to the curve and is above the x-axis, so only this one is going to be above the x-axis. And we're done. It's going to be y is equal to two.

How do you find the vertical tangent line of a derivative?

How to Find the Vertical Tangent.

Find the derivative of the function. The derivative (dy/dx) will give you the gradient (slope) of the curve..

Find a value of x that makes dy/dx infinite; you're looking for an infinite slope, so the vertical tangent of the curve is a vertical line at this value of x..

What is the equation of a vertical tangent line?

The curve y = f(x) has a vertical tangent line at the point (a, f(a)) if (i) f(x) is a continuous at x = a. of f(x), the limit should be an appropriate side limit. See Example 1).