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Worked example: Tangent to the graph of 1/x

A practice problem for finding the y-intercept of a tangent line to f(x) = 1/x. Created by Sal Khan.

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  • leaf green style avatar for user sriharicitizen
    Why is y = 1/k and x = k ?

    How is that possible ?
    (7 votes)
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    • leafers ultimate style avatar for user KrisSKing
      That's how the function is defined. At the beginning of the video Sal said we are dealing with the function y = 1/x. The specific question we are answering is what is the y-intercept when x = k. When x = k, y is equal to 1/k.
      (27 votes)
  • marcimus pink style avatar for user Maria Pia
    At , why is the derivative of the curve the same as the slope of its tangent line?
    (0 votes)
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  • old spice man green style avatar for user VIVEK RAO ANAND
    What is Local linearization?
    (4 votes)
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  • blobby green style avatar for user ericswann97
    what about tangent lines using a circle ?
    (1 vote)
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  • blobby green style avatar for user Josiah Cornett
    This is kind of the same thing as what he does in this video but it is sort of backwards and I'm having trouble with it. If you have the equation of a curve and are asked to find the points where the tangent line has a y-intercept of 5, how do you go about that?
    (3 votes)
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    • piceratops ultimate style avatar for user Just Keith
      Let us consider the facts we know about tangent lines:
      1. They must intersect the function at the point of tangency.
      2. They must have the same slope as does the function at the point of tangency.

      Using the point-slope form of the line, this means:
      y-y₁ = m(x-x₁)
      Where (x₁, y₁) is the point of tangency.
      Now, we need to replace m with the first derivative of the curve, since the first derivative can be interpreted as the slope of the curve (as a function instead of a constant).
      y-y₁ = (f')(x-x₁)
      where f' is the first derivative of the function.
      We are now given an additional piece of information, the y=intercept of the tangent line. in other words, the tangent line goes through the point (0, 5). We plug that in for x and y:

      5-y₁ = (f')(-x₁)

      Note: we would use the original function (not its derivative) to give us enough information to solve for two unknowns. We plug in the point of tangency (x₁,y₁) into the original function. This would give us two equations and two unknowns, which would allow us to solve for (x₁,y₁) (there may be multiple solutions).
      (3 votes)
  • blobby green style avatar for user Muneeb Saadi
    The slope of a line can be written as tan = y/x. if at k f(x) is 1/k then tan = (1/k)/k = 1/(k^2) but when derivative is taken it gives us -1/(k^2) why is that?
    (2 votes)
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    • marcimus pink style avatar for user Alex Tran
      Notice that the slope of the line you are finding gives the slope from the origin to (k, f(k)) for some some x = k. But the derivative of f(x) at x=k gives the slope exactly at x =k. These slope from the origin is a secant line, and the second is an instantaneous slope. Thus, their values are not the same in this example.
      (3 votes)
  • leafers seedling style avatar for user Ddayman13
    Are there more examples? I don't understand what he did to make x to the -1 turn into a -1 times x to the -2
    (1 vote)
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  • marcimus pink style avatar for user Kelly C
    at 2.40, why is the exponent of -1 in f(x) became -2 in f ' (x)?
    (1 vote)
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  • leaf green style avatar for user Abdo Reda
    I have a silly question, In the derivative introduction videos Sal said that for a function to be differentiable it must be continuous, Defined, has no vertical asymptotes and the limit of the derivative must exist if there are any other rules please tell me
    my question is how can a function like 1/x not be continuous and has a vertical asymptote still has a derivative of -x^-2 ?
    (1 vote)
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  • leaf green style avatar for user Ahmed
    Why is the y-value 1/k ?
    (1 vote)
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    • leafers ultimate style avatar for user Tim Lindsley
      Remember that the function is defined as f(x) = 1/x, where x does not equal 0. So if f(x) = 1/x, then any number we plug into the function is defined as 1 over that number.

      So if we said that x = 1, then f(1) = 1/1. If x = 2, then f(2) = 1/2. If x = pi, then f(pi) = 1/pi. And so on for any number that is not zero. In this example, x = k (k is an arbitrary constant), and therefore f(k) = 1/k.

      Now, keep in mind that f(x) is just a fancy way of saying y. We can define our x,y ordered pairs in terms of (x,y). Or, we can define them in terms of (x,f(x)). It's the same thing. But here, we know from the problem that x = k (note that Sal double-underlined that fact). So we can substitue k for x, which yields (k,f(k)). Since we know that the rule for f(x) = 1/x, and f(x) = y, and x = k, we know that f(x) = 1/x is the same as saying for a constant, k, f(k) = 1/k or as saying y = 1/k.
      (2 votes)

Video transcript

In terms of k, where k does not equal 0, what is the y-intercept of the line tangent to the curve f of x is equal to 1/x at the point of the curve where x is equal to k? So let's just think about what they're asking. So if I were to draw myself-- let me draw some quick axes right over here. So that's my y-axis. This is my x-axis right over here. And the graph f of x is equal to 1/x would look something like this. So it looks something like this. So it kind of spikes up there, and then it comes down, and then it goes like this. And I'm just doing a rough approximation of it. So it would look something like that. And then on the negative side, it looks something like this. So this is my hand-drawn version of roughly what this graph looks like. So this right over here is f of x is equal to 1/x. Now, we are concerning ourselves with the point x equals k. So let's say-- I mean, it could be anything that's non-zero, but let's just say that this is k right over here. So that is the point k 1/k. We can visualize the line tangent to the curve there. So it might look something like this. And we need to figure out its y-intercept. Where does it intercept the y-axis? So we need to figure out this point right over here. Well, the best way to do it, if we can figure out the slope of the tangent line here, the slope of the tangent line is just the derivative of the line at that point. If we could figure out the slope of the tangent line, we already know that line contains the point. Let me do this in a different color. We know it contains the point k comma 1/k. So if we know its slope, we know what point it contains, we can figure out what its y-intercept is. So the first step is just, well, what's the slope of the tangent line? Well, to figure out the slope of the tangent line, let's take the derivative. So if we write f of x, instead writing it as 1/x, I'll write it as x to the negative 1 power. That makes it a little bit more obvious that we're about to use the power rule here. So the derivative of f at any point x is going to be equal to-- well, it's going to be the exponent here is negative 1. So negative 1 times x to the-- now we decrement the exponent to the negative 2 power. Or I could say it's negative x to the negative 2. Now, what we care about is the slope when x equals k. So f prime of k is going to be equal to negative k to the negative 2 power. Or another way of thinking about it, this is equal to negative 1 over k squared. So this right over here is the slope of the tangent line at that point. Now, let's just think about what the equation of the tangent line is. And we could think about it in slope-intercept form. So we know the equation of a line in slope-intercept form is y is equal to mx plus b, where m is the slope and b is the y-intercept. So if we can get it in this form, then we know our answer. We know what the y-intercept is going to be. It's going to be b. So let's think about it a little bit. This equation, so we could say y is equal to our m, our slope of the tangent line, when x is equal to k, we just figure out to be this business. It equals this thing right over here. So let me write that in blue. Negative 1 over k squared times x plus b. So how do we solve for b? Well, we know what y is when x is equal to k. And so we can use that to solve for b. We know that y is equal to 1/k when x is equal to k. So this is going to be equal to negative 1-- that's not the same color. Negative 1 over k squared times k plus b. Now, what does this simplify to? See, k over k squared is the same thing as 1/k, so this is going to be negative 1/k. So this part, all of this simplifies to negative 1/k. So how do we solve for b? Well, we could just add 1/k to both sides and we are left with-- if you add 1/k here-- actually, let me just do that. Plus 1/k, left-hand side, you're left with 2/k, and on the right-hand side, you're just left with b, is equal to b. So we're done. The y-intercept of the line tangent to the curve when x equals k is going to be 2/k. If we wanted the equation of the line, well, we've done all the work. Let's write it out. It'll be satisfying. It's going to be y is equal to negative 1/k squared x plus our y-intercept, plus 2/k. And we're done.