We want to find the 35th and 82nd derivative
of f of x equals sine x.
To do this we'll find several derivatives,
notice a pattern that will allow us
to find the 35th and 82nd derivative.
So if we know f of x equals sine x,
then the first derivative f prime of x is equal
to cosine x.
The second derivative is equal to the derivative
of cosine x which is equal to negative sine x.
The third derivative is equal to the derivative
of negative sine x which equals negative cosine x.
And now when we find the fourth derivative,
we use this notation here where we put a four
in parenthesis here in the exponent position.
So this is the fourth derivative of f of x
which equals the derivative of negative cosine x
which equals negative negative sine x or positive sine x.
So notice here, the fourth derivative returns us back
to the original function f of x.
Which means from this point, our derivatives are going
to repeat.
For example, if we want to find the fifth derivative,
this is equal to the derivative of the fourth derivative
or the derivative of sine x which is the same
as our first derivative which equals cosine x.
And therefore the sixth derivative is equal
to the derivative of cosine x which is the same
as the second derivative so we have negative sine x
for our sixth derivative.
The seventh derivative would be equal to the derivative
of negative sine x which is the same
as the third derivative.
So the seventh derivative is negative cosine x
and the eighth derivative is equal to the derivative
of negative cosine x which is the same
as the fourth derivative sine x.
Now that we've noticed a pattern here,
that said we can generalize the pattern
to find the 35th and 82nd derivative.
Notice how if we're looking for a derivative
that would be a multiple of four.
The derivative would always be equal to sine x.
So we can say the four nth derivative of f of x
is always going to equal sine x.
As long as n is a positive integer.
Now let's take a look at this row here.
If we're looking for the first, fifth,
the next would be the ninth, thirteenth,
and so on derivative.
The derivative would always be equal to cosine x
which means if the derivative is in the form
of four n plus one, the derivative is always going
to equal cosine x.
Notice how the first, fifth, next would be the ninth
and 13th derivative would always fit the form
of four n plus one.
Next notice how the second, sixth, the next would be
the 10th then the 14th derivative
would all be negative sine x which means
the four n plus two derivative of f of x
is always equal to negative sine x.
And then finally we have the third, the seventh,
if we add four we'd be at the 11th derivative
then the 15th derivative and so on would always
be negative cosine x.
So if we're looking for the four n plus three derivative
of f of x, it'll always equal negative cosine x.
Now that we've determined the pattern of the derivatives,
in order to find any derivative, we just need to determine
which form the derivative fits.
So for example, in order to find the 35th derivative
of f of x, we're going to determine whether 35 fits the form
of four n plus one, four n plus two, four n plus three,
or four n.
One way to do this would be to take 35 and divide by four.
So if we take 35 and divide by four.
There are eight fours in 35.
Four times eight is 32 and we subtract.
We have a remainder of three.
Which means 35 is equal to four times eight plus three
which means 35 fits the form of four n plus three.
And, therefore, the 35th derivative is equal
to negative cosine x.
And now if we want to find the 82nd derivative,
we're going to determine whether 82 fits the form
of four n plus one, four n plus two,
four n plus three, or four n.
So again if we need to, we can take 82
and divide by four.
This is pretty straightforward.
There are 20 fours in 82.
Four times 20 is 80 with a remainder of two.
Which means 82 is equal to four times 20 plus two.
So 82, of course, fits the form of four n plus two
which means the 82nd derivative would be equal
to negative sine x.
So once we've noticed this pattern,
it's a pretty easy process to find any derivative
of f of x equals sine x.
We could use a similar strategy
to find higher order derivatives of f of x equals cosine x.
I hope you found this helpful.