## anonymous 4 years ago Find the derivative of the function-using the chain rule. k(x)= x^2 sec(1/x)

1. anonymous

need the product rule as well

2. anonymous

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

3. anonymous

okay

4. anonymous

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

5. anonymous

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

6. anonymous

yes

7. anonymous

okay,

8. anonymous

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

9. anonymous

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

10. anonymous

on account of the $$x^2$$ cancel

11. anonymous

okay,

12. anonymous

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

13. anonymous

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

14. anonymous

oh okay

15. anonymous

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

16. anonymous

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