A question on derivatives of leibniz

[transgalactic]

[SOLVED] a question on derivatives of leibniz

find the 50th derivetive of the function
f(x)=(x^2 * sin x)


i don't know how this stuff work
can you please show how to solve this question step by step

 

[bob1182006]

find the first few derivatives and look for a pattern, is it expanding, is the power of x increasing or decreasing? and how is sin changing between sin/cos i.e. even/odd derivatives you have +sin or -sin? or maybe +/- cos?

 

[transgalactic]

there is a formula of leibniz to get the answer
i hope some one explain to me step by step
how do i solve this question using this formula

solving this questin using leinbniz formula
?

 

[dynamicsolo]

i don't know how this stuff work
can you please show how to solve this question step by step

The rules of the forum require that you should some work first. People who are helping students here are not here to solve problems for those students.

If you are going to use a "formula of Leibniz", you need to show what that is and also show some attempt to use it. However, unless the problem specifically requires you to use a formula, you will find it easier to use bob1182006's suggestion of looking for a pattern. What is f'(x)? What is f''(x)?

 

[transgalactic]

i tried to solve it like this:

x^2*sinx +(50C1)2X*cosx - (50C2)*2*sinx
this way gives me a close answer but its still wrong
some were i did a mistake

 

[transgalactic]

i undrstood from the book that they are doing derivatives
of the lastobject till one of them turns to be only a number

but i don't think i got it right
help
?

 

[bob1182006]

usually you would do enough derivatives until you get just 0 if it's a polynomial.

but since you have sinx here which changes to cosx which changes to -sinx etc...

you need to find a pattern between the expansion of the derivatives.

Take say the first 3-4 derivatives and try to find a pattern, does every derivative contain sinx with some coeffieint etc.. so you can just find that pattern and find the 50th derivative that way.

 

[dynamicsolo]

i tried to solve it like this:

x^2*sinx +(50C1)2X*cosx - (50C2)*2*sinx
this way gives me a close answer but its still wrong
some were i did a mistake

I really think you're going to be unhappy trying to do this. You don't want to find all 50 derivatives at once. (In fact, I suspect you were asked to take the 50th derivative exactly to discourage you from using the formula.)

What bob1182006 (and I) are saying is to work out just a few higher derivatives until you can see what the pattern looks like, then suggest a general result for the n th derivative, and finally set n = 50.

 

[transgalactic]

the books showed a certain way to solve it

i added the example and the formula in the file

they gave me an example

but i didnt understand completely how this formula works

could you solve the question of
finding the 50th derivative of this function

f(x)=(x^2 *sin x)


??

 

[HallsofIvy]

It's really just the binomial formula:
$$(a+ b)^n= \sum_{i=0}^n _nC_i a^i b^{n-i}$$
the binomial coefficient is there because you are adding all the possible ways of "ordering" i "a"s and n-i "b"s.

Repeatedly differentiating a product gives a similar thing for exactly the same reasons:
[tex]\frac{d^n fg}{dx^n}= \sum_{i=0}^n _nC_i \frac{d^i f}{dx^i}{\frac{d^{n-i}g}{dx^{n-i}}[/itex]

For n= 50, that can have up to 51 terms. Fortunately, as you observe, the third derivative of x² is 0 so, taking f(x)= x², f'(x)= 2x, f"(x)= 2, and all other derivatives are 0. The formula becomes
$$_{50}C_0 x^2\frac{d^{50} sin(x)}{dx^{50}}+ _{50}C_1 (2x)\frac{d^{49}sin(x)}{dx^{49}}+ _{50}C_2 (2)\frac{d^{48}sin(x)}{dx^{48}}$$
So the only "problem" left is finding those derivatives of sin(x)- and that should be easy. (The derivatives of sine and cosine have "period" 4.)

Be careful of the signs- that's where your previous attempt is wrong.

 

[transgalactic]

thanks you very much
i finaly got his stuff all together