Question

need help on the attachment

Answer 1

are you looking for the derivative of
\[\tan(x)(2+\csc(x))\]?

Answer 2

hard to read, but if so forget the product rule and multiply out. you get
\[2\tan(x)+\sec(x)\] and then take the derivative term by term

Answer 3

what is this?

Answer 4

derivative of Tan x(2+csc x)

Answer 5

got it

Answer 6

turn it in to
\[2\tan(x)+\sec(x)\] first

Answer 7

then the derivative of tangent is secant squared, and the derivative of secant is secant tangent

Answer 8

so right away you get
\[2\sec^2(x)+\sec(x)\tan(x)\]

Answer 9

ok

Answer 10

but its doesn't match one of the answer choices

Answer 11

ok i see that so it is trig identity time. the calc is over

Answer 12

ok, lol

Answer 13

factor out a
\[\sec^2(x)\]

Answer 14

how can you factor out sec^2(x) when the second term has only one sec (x) with tanx

Answer 15

you get
\[\sec^2(x)(2+\frac{1}{\sec(x)}\tan(x))\]
\[\sec^2(x)(2+\cos(x)\tan(x))\]
\[\sec^2(x)(2+\sin(x))\]

Answer 16

good question i thought you might ask it

Answer 17

great response, lol

Answer 18

you factor it out by taking the reciprocal of secant, which is cosine

Answer 19

oh, ok so its all about manipulation till this point

Answer 20

all these things that are not sine and cosine are just combinations of them. so it is confusing, but they have no life of there own

Answer 21

i mean you don't even have secant on your calculator right?

Answer 22

no, but I do have cosine which you can just 1/cosx=secant

Answer 23

exactly my point. you can dispense with it entirely. it is just something made up to confuse you

Answer 24

and your next good question might be "how did you know to do that?"to which i would answer "well i saw the answers had a sec^2(x) factored out, so i factored it out"

Answer 25

oh, ok so really the we just simplify the function because we already took the derivative in the first part

Answer 26

right all the steps except the "take the derivative" step (only one step) was manipulating trig identities via algebra

Answer 27

well thanks so much satellite!

Answer 28

yw

Answer 29

forgot to ask where did the cosine go at the end when it was pair with tangent

Answer 30

????