# What It Means To Be 'Differentiable'

Coming up next: Using Limits to Calculate the Derivative

### You're on a roll. Keep up the good work!

Replay
Your next lesson will play in 10 seconds
• 0:06 Review of Derivatives
• 0:39 Velocity of a Jet
• 1:50 Velocity of a UFO
• 2:27 Velocity of a Superjet
• 3:36 Differentiable

Want to watch this again later?

Timeline
Autoplay
Autoplay
Speed

#### Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Eric Garneau
Lots of jets can go from zero to 300 mph quickly, but super-jets can do this instantaneously. In this lesson, learn what that means for differentiability.

## Review of Derivatives

Let's review really fast. The rate of change equals the velocity when we're thinking about it in terms of motion. The velocity and the rate of change are the slope of some position as a function of time. The instantaneous rate of change is your tangent to that line at any given point, and it is also known as the derivative. It's exactly how fast you're going at a single point in time.

## Velocity of a Jet

So what does this mean for a UFO, a jet and a superjet? Specifically, does each one of them always have a derivative, or a rate of change or a velocity? First, let's consider the jet. If I look at the jet's altitude as a function of time, I have a smooth graph. The jet takes off, increases its altitude and levels out at some really high-up altitude. At any point in time that jet has a velocity, and I can see that on the graph of h as a function of t. Initially the velocity is zero, my instantaneous velocity is zero, the tangent to this line is zero. Then the jet's gonna slowly start increasing its horizontal velocity, and the tangent to this curve has a higher slope. When it reaches the altitude that it's trying to get to, its horizontal velocity again goes to zero, and you see that up here. But at every point in time it has some velocity, even if that velocity is zero.

## Velocity of a UFO

Now let's compare this to a UFO. A UFO might be hanging out near some crops, and then all of a sudden it might jump to a higher altitude. Now, at this jump it has no velocity. I can't calculate an average velocity and have that average velocity converge on some tangent. It just doesn't exist. It doesn't exist because there's a discontinuous position. The graph h as a function of t has a discontinuity in it.

To unlock this lesson you must be a Study.com Member.

### Register to view this lesson

Are you a student or a teacher?

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.
Back
What teachers are saying about Study.com

### Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.