Derivative?

Question

Find the derivative.

(sin(5x))^10

Could you also explain how you got your answer? Thanks!

Answer 1

Use the chain rule:
First use the power rule on the outmost function, then take the derivative of the sin function, then take the derivative of the 5x

10(sin(5x))^9 * cos(5x) * 5 =

50(sin(5x))^9 * cos(5x)

Answer 2

10(sin(5x))^9 * (cos(5x) *5
so simplified= 50(sin(5x))^9 * cos(5x)

You will use the chain rule and work from the outside in. I'm guessing you now how to do basic derivatives. The most "outside" thing is the tenth power. Pretend (sin(5x)) is just "x" for now so you have x^10. Find the derivative= 10x^9. Substitute the (sin(5x)) back in and you have 10(sin(5x))^9 which is the first part. Now take the derivative of the next thing in which is the sin(5x). The derivative of sin x is cos x so you get (cos(5x). Finally the most inside is the 5x whose derivative is 5. Now multiply all the steps together and you get 10(sin(5x))^9 * (cos(5x) * 5 and simplify.

What was your main problem? i'm guessing the derivative of sin x?

Answer 3

I believe the answer is 50(sin(5x))^9 (cox(5x))
So first you get the 10 from (sin(5x))^10
and multiply it in front of the sin(5x), and also make the the sin(5x) to the ninth power
At this step, you should have
10(sin(5x)^9
Next, you have to follow the chain rule so you multiply the 10(sin(5x)^9 by the derivative of sin(5x). The derivative of sin(5x) is cos(5x).
Now you should have
10(sin(5x)^9 cos(5x)
It does not end here though, because you have to follow the chain rule one more time by multiplying the derivative of 5x from cos(5x). The x has a power of one, so just like any other derivative multiply x by its power (which is 1) and then subtract one from its power. This should leave you with 1, because x to the 0th power is 1. Back to the problem, we have to remember there was also a 5 in the 5x. So, 5 times 1 is 5 and we mulptiply that to our long derivative from before.
The result should be
10(sin(5x))^9 (cox(5x))5
which equals
50(sin(5x))^9 (cox(5x))

Answer 4

10(sin5x)^9 (5) (cos5x) = 50(cos5x)(sin5x)^9

You take the derivative of the power. Drop the 10 down and subtract 1 from the power. Then, take the derivative of the inside function (sin5x). This is cos5x. It is a composite function, so take the derivative of 5x, which is five. Multiply it all.

Answer 5

i was just doin math :'(.. omg i have a midterm tomm about derivatives.. i am not fully sure of the answer but nywayz:
50((sin(5x))^9)*cos5x

first get the derivative of the "power 10" and put it aside and minus one.. so it becomes (to the power 9). and keep finidng the derivate from outside in... ;) Good luck girl.

Answer 6

( (sin(5x))^10 ) ' = 10 * ( (sin (5x)) ^ 9) * (sin5x)' ... and (sin(5x))' = cos (5x ) * 5 (because (5x)') ...

so.. your answer should be 50* sin(5x)^9 * cos(5x)... there are some rules about derivative you'll fine in any math book.

Answer 7

Let u = sin 5x
du/dx = 5cos5xdx
dy/dx = 10u^9du/dx
dydx = 10(sin5x)^9 *5cos 5x
dy/dx = 50 cos 5x (sin x)^9

Answer 8

http://www.solvemymath.com/online_math_calculator/calculus/derivative_calculator/index.php
use this site it will help you get an answer and then you could check you own answer to see if it is right. it also sort of explains what is done.