Question

Find the derivative of the function-using the chain rule. k(x)= x^2 sec(1/x)

Answer 1

need the product rule as well

Answer 2

start with
\[2x\sec(\frac{1}{x})+x^2\frac{d}{dx}\sec(\frac{1}{x})\] second part requires chain rule

Answer 3

okay

Answer 4

the one that you have done is by using the product rule?

Answer 5

the derivative of secant is secant tangent, and the derivative of \(\frac{1}{x}\) is \(-\frac{1}{x^2}\)

Answer 6

yes

Answer 7

okay,

Answer 8

so the whole thing is
\[2x\sec(\frac{1}{x}+x^2\sec(\frac{1}{x})\tan(\frac{1}{x})\times (-\frac{1}{x^2})\]

Answer 9

we can clean it up a bit as
\[2x\sec(\frac{1}{x})-\sec(\frac{1}{x})\tan(\frac{1}{x})\]

Answer 10

on account of the \(x^2\) cancel

Answer 11

okay,

Answer 12

If anytime i get a problem like this, do I have to use the product rule first and then continue with the chain rule?

Answer 13

well it is not really a matter of "what goes first" you have to use the rules as you need them
\(x^2\sec(\frac{1}{x})\) is a product so you need the product rule for sure
also \(\sec(\frac{1}{x})\) is a composite function, so you must use the chain rule when you take the derivative

Answer 14

oh okay

Answer 15

just like if you have a quotient, you have to use the quotient rule, but if the numerator is a product, you will need the product rule for that one and if the denominator is a composite function you will need the chain rule for it

use whatever rules you need to get the derivative

Answer 16

i have one more question/problem.would you be willing to help me out ?