Question

how to find nth derivative of sin x

Answer 1

Do you know the derivative of sin x?

Answer 2

You will also need to know the derivative of cos x, do you know these derivatives?

Answer 3

what is nth derivation of sin

Answer 4

To figure that out you first need to know the derivative of sin x? Do you know this?

Answer 5

cos

Answer 6

That's right, now what's the derivative of cos x?

Answer 7

-sin

Answer 8

Yup, so the first derivative of sin x is cos x, the second derivative is -sin x, the third derivative will be -cos x and then the fourth will be sin x again.

Answer 9

but nth one

Answer 10

So now, to get the nth derivative, we must try to see a pattern between n and one of these functions (sin x, cos x, -sin x and -cos x)

Answer 11

The fourth derivative of sin x was itself, so what do you think the eight derivative is?

Answer 12

sin

Answer 13

Yep, if we do the derivative of each of the functions four times we get back to where we started. An example is the 765th derivative of sin(x) is the same as it's 1st derivative cos(x), because 765-4*191=1. And we can take away four any number of times since we get back, do you understand?

Answer 14

sin

Answer 15

So the n'th derivative depends on whether n gives rest 0, 1, 2 or 3 with division by 4.

Answer 16

number division by4 also division by2

Answer 17

So one way to write all this in a mathematical form would be:
\( \frac{d^n}{d^nx}(\sin x)=(-1)^{n/2}\sin x \) - If n is even.
\( \frac{d^n}{d^nx}(\sin x)=(-1)^{n-1/2}\cos x \) - If n is odd.

Answer 18

But these expressions are pretty clumsy, the idea is more important.

Answer 19

ok

Answer 20

So, if you understood you should be able to do the 1035th derivative of sin(x), if you know that 1035 gives rest 3 with division by 4.