Question

what is the 20th derivative of y=sin(2x)?

Answer 1

for these type of problems you need to find the pattern in the first few derivatives to make a guess for the nth derivative
or in this case the 20th derivative

Answer 2

so f(x)=sin(2x)
and f'(x)=?

Answer 3

2cos(2x)

Answer 4

\[f(x)=\sin(2x) \\ f^{(1)}(x)=2\cos(2x) \\ f^{(2)}(x)=-4\sin(2x)\]
what is the 3rd derivative?

Answer 5

would it be -8cos(2x)?

Answer 6

\[f(x)=\sin(2x)=2^0\sin(2x) \\ f^{(1)}(x)=2\cos(2x)=2^1 \cos(2x) \\ f^{(2)}(x)=-4\sin(2x) =- 2^2 \sin(2x)\\ f^{(3)}(x)=-8 \cos(2x)=-2^3 \cos(2x) \]

Answer 7

so you should a couple of things in pattern

Answer 8

see*

Answer 9

It looks like every two will be positive, right?
and also you notice we have the sin and cos switching with every term.
And what else do you notice?

Answer 10

the degree of 2 is increasing

Answer 11

so we know that the power of 2 is going to be what for the 20th derivative?

Answer 12

for the first derivative we had 2^1
second we had 2^2
third we had 2^3
so
20th we had 2^?

Answer 13

20?

Answer 14

right

Answer 15

ohhhhh i seee

Answer 16

and lets determine if we need cos or sin

Answer 17

look at the even derivatives and the odd derivatives

Answer 18

20 is even what do you notice for all the even derivatives so far

Answer 19

they have been sin right?

Answer 20

so the 20th derivative would be correct to assume we will have sin

Answer 21

\[f^{(20)}(x)= (\text{ is this plus or minus) }2^{20}\sin(2x)\]

Answer 22

we almost are done with this

Answer 23

zero derivative=pos
1st derivative=pos
2nd derivative=neg
3rd derivative=neg

so 20 divided by 4=5 remainder 0
the remainder 0 tells us that we have pos
since the pattern repeats itself

Answer 24

so 2^(20)*sin(2x)

Answer 25

This makes so much sense ! THANK YOU

Answer 26

if we had to find the 22nd derivative this would be
\[f^{(22)}(x)=-2^{22}\sin(2x)\]
22/4=5 remainder 2
and recall the 2nd derivative=neg
so that is how we got the 22nd derivative was going to be negative

Answer 27

but anyways I was just showing you one more for fun

Answer 28

did you want to see if you could find the 46th derivative?

Answer 29

even derivative we have sin
odd derivative we have cos
-
we had 2^(the number of derivative we were on)
-
follow the pattern
zero derivative=pos
first derivative=pos
second derivative=neg
third derivative=neg
to find the sign (neg or pos sign that is)

Answer 30

and no problem :)