Question

Can someone work me through this step by step?

Find dy/dt of y=ln(sec(3t)+tan(7t))

Would be greatly appreciated!

Answer 1

You just need to use chain rule a lot.

So the first step is

\[dy/dt = (d/dx[\sec(3t) + \tan(7t)]) / ((\sec(3t) + \tan(7t))\]

Answer 2

let u = sec(3t) + tan(7t)

d/dx(ln u ) = u'/u

Answer 3

combines derivative of natural log and chain rule

Answer 4

Now to derive sec(3t) + tan(7t)

= 3[sec(3t)tan(3t)] + 7sec^2(7t)

Answer 5

You just plug it back in...

(3sec(3t)tan(3t) + 7sec^2(7t)) / (sec(3t) + tan(7t))

Answer 6

which should be your answer. If you have answers, feel free to ask :)

Answer 7

Thanks! It might take me a second I'm still trying to learn this stuff so I'm going to go through it step by step, but if I have a question I'll add it here if you're still on! Thanks :)

Answer 8

You're very welcome

Answer 9

Can you explain why (lnu)' = u'/u? Is that just a rule that always holds true? I learned that (lnx) = 1/(x), so there's my first area of confusion hahah wouldn't it be the same for u? Sorry if this is obvious!

Answer 10

(lnx)' = 1/(x) ****

Answer 11

I do see that working out u'/u for x = 1/x so that makes sense. But for more complicated problems it's appropriate to use x'/x for lnx?

Answer 12

It's true that d/dx(ln(x)) is always 1/x

However, pretend that u is any rational function.

The derivative of a function of a function is in the form
d/dx[f(g(a))] = f'(g(a)) + g'(a)

This is the chain rule.

So d/dx[ln(x)] = 1/x

But if it's not just x and a function, we also have to multiply it by the derivative of the inside

Answer 13

This is just how I learned the chain rule. Don't let the u confuse you

Answer 14

EDIT: I meant
The derivative of a function of a function is in the form
d/dx[f(g(a))] = f'(g(a)) * g'(a)

Answer 15

Okay thanks a ton! That helps!

Answer 16

If you haven't learned the chain rule in depth, I strongly recommend watching a few Youtube videos to help teach you because they can get really hard to see

Answer 17

Yeah I think I need that! Going to do that now.

Answer 18

Tiene suerte

Answer 19

Awesome! Understand this completely thanks once more. You should get paid for this, if you don't already!