Differentiable means that the derivativeexists ..
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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! How to advertise your ebay store. x2 + 6x is differentiable.
.. and it must exist for every value in the function's domain.
DomainIn its simplest form the domain is |
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.
But what about this:
Example: The function f(x) = |x| (absolute value):
|x| looks like this: |
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! How to advertise your ebay store. x2 + 6x is differentiable.
.. and it must exist for every value in the function's domain.
DomainIn its simplest form the domain is |
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.
But what about this:
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?
Testing
We can test any value 'c' by finding if the limit exists:
2.3 Differentiabilityap Calculus Definition
limh→0f(c+h) − f(c)h
Example (continued)
Let's calculate the limit for |x| at the value 0:
The limit does not exist! To see why, let's compare left and right side limits:
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?
2.3 Differentiabilityap Calculus Notes
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).
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.
Enter an expression and the variable to differentiate with respect to. Then click the Differentiate button.
Finding the derivative of
involves computing the following limit:
To put it mildly, this calculation would be unpleasant. We would like to find ways to compute derivatives without explicitly using the definition of the derivative as the limit of a difference quotient. A useful preliminary result is the following:
Derivative of a Constant
lf c is any real number and if f(x) = c for all x, then f ' (x) = 0 for all x . That is, the derivative of a constant function is the zero function.
It is easy to see this geometrically. Referring to Figure 1, we see that the graph of the constant function f(x) = c is a horizontal line. Since a horizontal line has slope 0, and the line is its own tangent, it follows that the slope of the tangent line is zero everywhere.
We next give a rule for differentiating f(x) = xn where n is any real number.Some of the following results have already been verified in the previous section, and the
others can be verified by using the definition of the derivative.
This pattern suggests the following general formula for powers of n where n is a positive integer.
Power Rule
In fact, the power rule is valid for any real number n and thus can be used to differentiate a variety of non-polynomial functions. The following example illustrates some applications of the power rule.
Example 1
Differentiate each of the following functions:
(a) Since f(x) = 5, f is a constant function; hence f '(x) = 0.
(b) With n = 15 in the power rule, f '(x) = 15x14
(c) Note that f(x) = x1/2 . Hence, with n = 1/2 in the power rule,
(d) Since f(x) = x-1, it follows from the power rule that f '(x) = -x-2 = -1/x2
The rule for differentiating constant functions and the power rule are explicit differentiation rules. The following rules tell us how to find derivatives of combinations of functions in terms of the derivatives of their constituent parts. In each case, we assume that f '(x) and g'(x) exist and A and B are constants.
The four rules listed above, together with the rule on differentiating constant functions and the power rule, provide us with techniques for differentiating any function that is expressible as a power or root of a quotient of polynomial functions. The next series of examples illustrates this. The linearity rule and the product rule will be justified at the end of the section; a proof of the extended power rule appears in the section on the chain rule.
Example 2 Let
Find f '(x).
Solution Using the linearity rule, we see that
Example 3 Let
Again using linearity,
f'(x) = a(x3)' + b(x2)' + c(x)' + (d)' = 3ax^2 + 2bx + c
Example 3 can be generalized as follows:
A polynomial of degree n has a derivative everywhere, and the derivative is a polynomial of degree (n - 1).
Example 4 Let
Find f '(x).
First we use the product rule, since f(x) is given as the product of x2 and x2 -x + 1: