Question

Finding the derivative of a problem.. (Problem is in the comments)

Answer 1

I'm not sure if you would use the chain rule for this?

Answer 2

g(t)=5

Answer 3

wait what?

Answer 4

yes - use chain rule

Answer 5

Canyou explain how you did that?

Answer 6

is the inner part πt and the outer part 5cos^2?

Answer 7

hold on - i've confused myself here
we can use the product rule as well
function = 5 cos pi t * cos pi t
inner function = pi t , outer = cos

Answer 8

d(cos(pi t)) / dt = - pi sin (pi t)

Answer 9

wait if you used the product rule wouldn't it become this:
5cos^2 π + -10sin (πt)

Answer 10

because you find the derivative of the second times the first plus the derivative of the first times the second?

Answer 11

right or you could take the 5 out first then differentiate whats in brackets

5 ( cos (pi t) * cos (pi t) )

Answer 12

so is my answer then
y'= 5πcos^2 -10sinπ t ?

Answer 13

I get it -10pi sin (pi t) cos (pi t)

Answer 14

how did you get that from what I did before?

Answer 15

which can be simplified to -5pi sin (2 pi t)

Answer 16

maybe theres a simpler way to do this...

Answer 17

I think I would use the product rule
\[ \frac{d}{dt} x^2 = 2 x \frac{d}{dt} x \]

here x is \( \cos(\pi t) \)

the 5 is a constant, which we can put "out front"
\[ 5 \frac{d}{dt} \cos(\pi t)^2 = 5\cdot 2 \cos(\pi t)\frac{d}{dt} \cos(\pi t) \\
=10 \cos(\pi t)\frac{d}{dt} \cos(\pi t)
\]
now you need to find the derivative of cos(pi*t)

Answer 18

**I think I would use the *power* rule

Answer 19

yes - good way to do it

Answer 20

is the derivate of cos pi t
-pi (sin)?

Answer 21

we know
\[\frac{d}{dt} \cos(x) = -\sin(x) \frac{d}{dt} x \]
here x is \( \pi t\)
\[ \frac{d}{dt} \cos(\pi t)= -\sin(x) \frac{d}{dt} \pi t \]
last step is the derivative of pi*t

Answer 22

yes derivative of cos pi t = - pi sin pi t

Answer 23

\[ \frac{d}{dt} \cos(\pi t) = - \sin(\pi t) \pi = -\pi\sin(\pi t) \]

Answer 24

putting it together
\[ -10 \pi \cos(\pi t) \sin(\pi t) \]

Answer 25

thankyou!

Answer 26

There is the trig identiy crw used, that let's you rewrite
2 cos(x) sin(x) as sin(2x)

You may or may not use it depending on the problem
so an alternate way to write the derivative is
\[ - 5 \pi \sin(2\pi t) \]

Answer 27

*cwrw238