Question

differentiate y = sec (sqrt x) tan (1/x)

Answer 1

\[y = \sec \sqrt{x} \tan (1/2)\]

Answer 2

tan(1/2) or tan(1/x)?

Answer 3

1/x. sorry

Answer 4

no worries!

Answer 5

so any idea to differentiate it? the answer is quite weird.

Answer 6

You want to use the product rule.
So the derivative of sec(sqrt(x)) times tan(1/x) plus the derivative of tan(1/x) times sec(sqrt(x))

Answer 7

yes. i'm using the product rule. but it seem complicated to me. can u show me?

Answer 8

okay so the derivative of sec(sqrt(x)) = sec(sqrt(x))*tan(sqrt(x))*(1/2*x^(-1/2))

Answer 9

\[\sec(\sqrt(x))\tan(\sqrt(x)) / (2\sqrt(x))\]

Answer 10

differentiate tan x get sec^2 x right?

Answer 11

yes but hold on one sec haha

Answer 12

then you multiply the derivative of sec(sqrt(x)) by tan(1/x) and you get \[\sec(\sqrt(x))\tan(\sqrt(x))\tan(1/x)/(2\sqrt(x))\]

Answer 13

then you add the second half of the product rule, the derivative of tan(1/x) times sec(sqrt(x))

Answer 14

the derivative of tan(x) is, as you said, sec^2(x).
however, you also need to apply the chain rule to get sec^2(1/x) times the derivative of 1/x, -1/x^2

Answer 15

What are you two owls doing here? Didn't I tell you no midnight owls doing math in my attic?

Answer 16

so, for the second half of the product rule you get \[\sec^2(1/x)*-1/(x^2)*\sec(\sqrt(x))\]

Answer 17

\[\sec \sqrt{x}(\sec ^2 1/x)(-x^-2)+ \tan 1/x (\sec \sqrt{x} \tan \sqrt{x})(1/2 x^-1/2)\]

Answer 18

i get this. but how to simplify it?

Answer 19

does it ask you to simplify?

Answer 20

yes. until the simplified term

Answer 21

\[(\sec \sqrt{x})\left[( \tan \sqrt{x}\tan (1/x))/2\sqrt{x})-(\sec^2 (1/x)) /x^2 \right]\]

Answer 22

this is the last question. i don't get it

Answer 23

sorry. the final answer.