# 2.3 Differentiabilityap Calculus

Differentiable means that the derivativeexists ..

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.

Posted by 3 years ago. Can someone help me with this question? (Ap Calc AB - Graphing/differentiability) Ap calculus. Differentiate any function with our calculus solver. Enter an expression and the variable to differentiate with respect to. Then click the Differentiate button. FREE-RESPONSE SOLUTIONS 2019 AB Question AB-2 (a) vtP is differentiable and therefore continuous on 0.3,2.8.Since 2.8 0.3 55 55 0 2.8 0.3 2.5 vvPP. 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.

### Example: is x2 + 6x differentiable?

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.

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

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

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

 x looks like this:

At x=0 it has a very pointy change!

Does the derivative exist at x=0?Format external hard drive for mac using windowslasopafs.

## Testing

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

limh→0f(c+h) − f(c)h

### Example (continued)

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

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'.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).

## 2.3 Differentiabilityap Calculus Test

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.

Differentiable Continuous

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

(1) Find the derivatives of the following functions using first principle.

(i) f(x) = 6Solution

(ii) f(x) = -4x + 7Solution

(iii) f(x) = -x2 + 2 Solution

(2) Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?

(i) f(x) = x - 1 Solution

(ii) f(x) = √(1 - x2) Solution

(3) Determine whether the following function is differentiable at the indicated values.

(i) f(x) = x x at x = 0 Solution

(ii) f(x) = x2 - 1 at x = 1 Solution

(iii) f(x) = x + x - 1 at x = 0, 1 Solution

(iv) f(x) = sin x at x = 0 Solution

(4) Show that the following functions are not differentiable at the indicated value of x.

(i)

Solution

(5) The graph of f is shown below. State with reasons that x values (the numbers), at which f is not differentiable.

Solution

(6) If f(x) = x + 100 + x2, test whether f'(-100) exists.

(7) Examine the differentiability of functions in R by drawing the diagrams.

(i) sin x Solution

## 2.3 Differentiabilityap Calculus Notes

(ii) cos x Solution

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

[email protected]

You can also visit the following web pages on different stuff in math.

 ALGEBRANegative exponents rulesCOMPETITIVE EXAMSAPTITUDE TESTS ONLINEACT MATH ONLINE TESTTRANSFORMATIONS OF FUNCTIONSORDER OF OPERATIONSWORKSHEETS TRIGONOMETRYTrigonometric identitiesMENSURATIONGEOMETRYCOORDINATE GEOMETRYCALCULATORSMATH FOR KIDSLIFE MATHEMATICSSYMMETRYCONVERSIONS

WORD PROBLEMS

HCF and LCM word problems

Word problems on simple equations

Word problems on linear equations

Trigonometry word problems

## 2.3 Differentiabilityap Calculus Calculator

Word problems on mixed fractrions ## 2.3 Differentiabilityap Calculus Definition

OTHER TOPICS

Ratio and proportion shortcuts

Converting repeating decimals in to fractions

SBI!