2.3 Differentiabilityap Calculus

AP Calculus AB - Worksheet 33 Derivatives of Inverse Trigonometric Functions Know the following Theorems. Find the derivative of y with respect to the appropriate variable. Lesson 2.6: Differentiability: Afunctionisdifferentiable at a point if it has a derivative there. In other words: The function f is differentiable at x if.

  1. 2.3 Differentiabilityap Calculus Definition
  2. 2.3 Differentiabilityap Calculus Notes
  3. 2.3 Differentiabilityap Calculus Calculator
  4. 2.3 Differentiabilityap Calculus Test

Differentiable means that the derivativeexists ..

Example: is x2 + 6x differentiable?

2.3 Differentiabilityap Calculus

Derivative rules tell us the derivative of x2 is 2x and the derivative of x is 1, so:

Its derivative is 2x + 6

So yes! x2 + 6x is differentiable.

.. and it must exist for every value in the function's domain.

Domain

In its simplest form the domain is
all the values that go into a function

Example (continued)

When not stated we assume that the domain is the Real Numbers.

Test

For x2 + 6x, its derivative of 2x + 6 exists for all Real Numbers.

So we are still safe: x2 + 6x is differentiable.

But what about this:

Example: The function f(x) = x (absolute value):

2.3 Differentiabilityap Calculus Definition

Differentiabilityap
x looks like this:

At x=0 it has a very pointy change!

Does the derivative exist at x=0?

Testing

2.3 Differentiabilityap Calculus Notes

We can test any value 'c' by finding if the limit exists:

limh→0f(c+h) − f(c)h

2.3 Differentiabilityap Calculus

Example (continued)

Let's calculate the limit for x at the value 0:

2.3
f(x) = x :limh→0 c+h − c h
Simplify:limh→0 h h

The limit does not exist! To see why, let's compare left and right side limits:

From Right Side:limh→0+ h h = +1

The limits are different on either side, so the limit does not exist.

So the function f(x) = x is not differentiable

A good way to picture this in your mind is to think:

As I zoom in, does the function tend to become a straight line?

The absolute value function stays pointy even when zoomed in.

Other Reasons

Here are a few more examples:

The Floor and Ceiling Functions are not differentiable at integer values, as there is a discontinuity at each jump. But they are differentiable elsewhere.

The Cube root function x(1/3)

Its derivative is (1/3)x−(2/3) (by the Power Rule)

At x=0 the derivative is undefined, so x(1/3) is not differentiable.

At x=0 the function is not defined so it makes no sense to ask if they are differentiable there.

To be differentiable at a certain point, the function must first of all be defined there!


As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is 'heading towards'. Camcamx serial mac lookup.

So it is not differentiable.

Different Domain

But we can change the domain!

Example: The function g(x) = x with Domain (0,+∞)

The domain is from but not including 0 onwards (all positive values).

Which IS differentiable.

And I am 'absolutely positive' about that :)

So the function g(x) = x with Domain (0,+∞) is differentiable.

We could also restrict the domain in other ways to avoid x=0 (such as all negative Real Numbers, all non-zero Real Numbers, etc).

Why Bother?

Because when a function is differentiable we can use all the power of calculus when working with it.

Continuous

When a function is differentiable it is also continuous.

2.3 Differentiabilityap Calculus Calculator

Differentiable Continuous

2.3 Differentiabilityap Calculus Test

But a function can be continuous but not differentiable. For example the absolute value function is actually continuous (though not differentiable) at x=0.