18.) Find the 65th derivative of sin(x) by finding the first few derivatives and observing the pattern that occurs. (sin(x))(65)=?

sin x goes to cos x which goes to - sin x which goes to - cos x and then the cycle restarts at sin x.

so there are 4 events in this cycle.

1) sin x

2) cos x

3) - sin x

4) -cos x

you want the 65th. you know 64 goes into 4, 16 times. so thats 16 complete cycles.

so the 65th would be the start again.

First derivative = cos(x)

Second derivative = -sin(x)

Third derivative = -cos(x)

Fourth derivative = sin(x)

See the pattern?

How many times does it repeat by the 65th derivative?

Did you at lest try their suggestion?

And again, *you* must learn by heart the derivative of the basic functions. We cant learn by heart for you.

So, whats the derivative of sin(x) ?

Then, the derivative of the above answer?

Then the derivative of that answer?

Etc... do it a few more times.

See the pattern?

Hopefully no one will spoil you the answer thereby depriving you from your personal enhancement; that would be very inconsiderate of them.