# Find the n'th derivative of these

• transgalactic
In summary: The derivative of |x|n+1 at x= 0 is not defined when x= 0, because the function is not continuous there. However, the derivative of |x|n+1 at x= 0 is defined for all other values of x.In summary, we discussed the derivatives of absolute value functions and how they are not continuous at x = 0. We also looked at the Makloren formula for approximating functions and how to take the nth derivative from it. Additionally, we explored the nth derivative of |x|^(n+1) and how to generalize the process for finding higher derivatives.
transgalactic
1=<k=<n
find $$f^{(n)} (x)$$ of:

the first :
$$f(x) = |x|^{n + 1}$$
the second is:
$$f(x) = |\sin x|^{n + 1}$$

What is the use of defining k?
Did you try calculating the first derivatives by hand and looking for a pattern?

what is the derivative of an absolute value function??

For you to solve an absolute value derivative you have to realize that there are 2 parts of the function: + and -. Therefore you take the derivative of both parts and set the regions where the derivative is true.

For example:
f(x)=|x|

The line y=-x exists from -infinity to 0 and the line y=x exists from 0 to +infinity. Then the derivative of f(x) is -1 on (-infinity, 0) and 1 on (0,infinity). Hope this helps!

in an ordinary function
the derivative from he left must equal the derivative from the right

so we can't do a derivative to |x|
because of that?? on one side i get 1 on the other i get -1
the same thing goes for the question i need to solve
??

Note that what you say is a pointwise statement.
Only for x = 0 you get different limits on both sides, which means that |x| is not differentiable at x = 0. This corresponds to the intuitive notion of differentiability, which |x| doesn't satisfy at x = 0 because it has a kink / sharp point there.
However, for x non-zero, the function is perfectly smooth and differentiable.

In fact, when you do analysis you start by defining what it means if f is differentiable at a point (see your other thread, where I gave the definition) and then for colloquial convenience define just "f is differentiable" to mean: "f is differentiable at all the points in the domain".

the makloren formula is
$$f(x)=\sum_{n}^{k}\frac{f^{(k)}}{k!}x^k+o(x^n)$$

how to take the n'th derivative from here??

where to put the formula in and get the n'th derivative??

this function
is for approximating so in order for me to get to the n'th members approximation
first i need to do manually one by one n times derivative
so its not helping me
??

for the second question:
$$(|\sin x|^{n}||\sin x|)^{(n)}=\sum_{k=0}^{n}C^k_n (|\sin x|^{n})^{(n-k)}|sinx|^{(k)}\\$$

what to do next??

Differntiating f(x)= |x|n+1 is easy. if x> 0 then f(x)= xn+1 what is the nth derivative of that? For example, if n= 3, (x4)'= 4x3, (x4)"= (4x3)'= 4(3)x2, (x4)'''= (4(3)x2)'= (4)(3)(2)x. Can you generalize that?

If x< 0, f(x)= -xn+1 whenever n+1 is odd.

The derivative of |x|n+1 at x= 0 is not defined when x= 0.

## What is the purpose of finding the n'th derivative?

The n'th derivative is an important mathematical concept used to determine the rate of change of a function. It is particularly useful in physics and engineering, where it can be used to model and analyze various systems.

## How do you find the n'th derivative of a function?

The n'th derivative of a function can be found by repeatedly taking the derivative of the function n times. Alternatively, it can also be calculated using the general formula for n'th derivatives, which involves using the n'th derivative rule and the power rule.

## What is the difference between the first derivative and the n'th derivative?

The first derivative represents the rate of change of a function at a specific point, while the n'th derivative represents the rate of change of the rate of change, and so on. In other words, the n'th derivative measures how the rate of change of a function changes as we move along the x-axis.

## Why is it important to know the n'th derivative of a function?

The n'th derivative is important because it provides valuable information about the behavior of a function, such as the maximum and minimum points, the inflection points, and the intervals of concavity and convexity. This information can be used to understand and analyze the behavior of complex systems.

## Can the n'th derivative be used to find the slope of a tangent line?

Yes, the n'th derivative can be used to find the slope of a tangent line at any point on a curve. The first derivative gives the slope of the tangent line, the second derivative gives the rate of change of the slope, and so on. This allows us to approximate the behavior of a function at a specific point.

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