Question

hey I need some chain rule help *advanced chain rule* f(x)=x^5sec(1/x)

answer is: -x^3sec(1/x)tan(1/x)+5x^4sec(1/x)
I need to knw how to get the answer. This is so I can study for my Ap Calc AB final on monday. PLEASE HELP!! Thanks.

Answer 1

You can apply quotient rule if the product rule is messy

Answer 2

\[f(x)=x^5\sec(1/x) \\ \\ f(x)=\frac{x^5}{\cos(1/x)}\]

Answer 3

i know but I am having trouble with the -x^3 part in the answer. I got everything else except for that

Answer 4

\[\LARGE f'(x)=\frac{\cos(1/x)5x^4+x^5\sin(1/x)(-1/x^2)}{\cos^2(1/x)}\]

Answer 5

Have you checked your signs?

Answer 6

i got the answer from the back of the book. This isnt for homework, I'm just really anxious about this problem. the only part of the problem that I didnt get right was the -x^3 part.

Answer 7

Hmm, that part is from
\[\Large f'(x)=\frac{\cos(1/x)5x^4\color{blue}{+x^5\sin(1/x)(-1/x^2)}}{\cos^2(1/x)}\]

Answer 8

im still not following you..Can you solve it using chain rule? because thats how the answer is expressed.

Answer 9

But this way will be easier, maybe this you'll know, you know quotient rule is,
\[f(x)=\frac{u}{v} \\ \\ \large f'(x)=\frac{vu'-\color{red}{u}\color{blue}{v'}}{v^2}\]
From red we have \(\color{red}{x^5}\)
From \(\color{blue}{blue}\) is the derivative of \(\cos(\frac{1}{x})\), which requires chain rule, chain rule is
\[\frac{dy}{dx}=\frac{dy}{du} \cdot \frac{du}{dx}\]so,
\[\frac{d}{dx}(\cos(\frac{1}{x}))=-\sin(\frac{1}{x})[\text{multiplied by the derivative of inside, or 1/x}\]
\[\frac{d}{dx}(\cos(\frac{1}{x}))=\color{blue}{-\sin(\frac{1}{x})(-\frac{1}{x^2})}\]
Combining red and blue we have
\[x^5\sin(\frac{1}{x})(\frac{1}{x^2}) \\ \\ =x^3\sin(\frac{1}{x})\]

Answer 10

maybe Im just an idiot, but i still dont see what you are saying. The sec(1/x) comes after the -x^3, so your above example is seeming irrelevant. I know how to use product and and quotient rule fairly well. I just dont understand how the chain rule was applied to this problem.

Answer 11

Chain rule is applied when
\[x^n\] when \(n \neq1\)

Answer 12

@Omari248 Sam convert sec (1/x) = 1/cos(1/x) to apply quotient rule

Answer 13

i know that. I have 75% of this problem correct, I just dont have the -x^3 that was shown in the answer. Ive tried this problem at least 5 times and have not seen how that shows up in the answer. i got:
5x^4sec(1/x)+sec(1/x)tan(1/x)

Answer 14

so your original one become \[\frac{x^5}{cos(\frac{1}{x)}}\]

Answer 15

look. Can one of you guys just solve the problem using chain rule ONLY. because now Im just getting confused.

Answer 16

alright thanks.

Answer 17

I wish you had used chain rule instead of quotient rule, but whatever. I guess Ill just get that one wrong on my final.

Answer 18

hey, that includes chain rule there, so, when you take (cos(1/x) ' , the part (1/x)' is called CHAIN RULE,my lord