Question

@zepdrix Find the derivative of the function.
f(t)=cos^6(pi t -14)

Answer 1

\[\large f(t)=\cos^6\left(\pi t-14\right)\]

Are you able to recognize what the outermost function is?
The notation we use for trig functions can make that a little difficult sometimes.
Writing it this way might help if you're having any trouble with that particular idea.\[\large f(t)=\left[\cos\left(\pi t-14\right)\right]^6\]

Answer 2

So to take the derivative, we will apply the power rule to the `outermost function`.
Then we'll multiply by the derivative of the inner function.

Answer 3

Power rule gives us this, right?
\[\large f'(t)=6\cos^5\left(\pi t-14\right) \color{royalblue}{\left[\cos(\pi t-14)\right]'}\]
Chain rule tells us to multiply by that blue term.
We still need to take the derivative of that blue term.

And confusion yet? :O

Answer 4

So far so good!

Answer 5

\[\large f'(t)=6\cos^5\left(\pi t-14\right) \color{orangered}{\left[-\sin(\pi t-14)\right]}\color{royalblue}{\left(\pi t-14\right)'}\]
Cosine gives us -sine right?
We then apply the chain rule once more, multiplying by the derivative of the inner inner function.

Answer 6

Mmkay

Answer 7

Why do we keep adding on (pi t -14)?

Answer 8

Ok from this step,
\[\large f'(t)=6\cos^5\left(\pi t-14\right) \color{royalblue}{\left[\cos(\pi t-14)\right]'}\]

The pi*t-14 is inside of the trig function.
When we apply the chain rule, mmm trying to think of the right way to explain this...
We're making a copy of the cosine function (which includes it's innards), and taking its derivative.

Answer 9

Confused by the next (pi*t-14) that shows up after this one?

Answer 10

I think I got it

Answer 11

The derivative of that last portion gives us,
\[\large f'(t)=6\cos^5\left(\pi t-14\right)\left[-\sin(\pi t-14)\right]\color{orangered}{\left(\pi\right)}\]

Answer 12

okay

Answer 13

Since multiplication is `commutative`, we can rearrange the order in which we multiply.
This is just to make it look a little cleaner.\[\large f'(t)=-6\pi \cos^5(\pi t-14)\sin(\pi t-14)\]

Answer 14

Chain rule is a little tricky to get used to.
Practice more of these!! :)

Answer 15

who that's the final answer! wow you are so helpful! And I know I've been having trouble with chain rule :(