Question

find dy/dx by implicit differentiation

Answer 1

\[e^ycosx=1+\sin(xy)\]

Answer 2

I am not sure how to differentiate sin(xy)

Answer 3

what is differentiation of sin(.) ??

Answer 4

cos(.)

Answer 5

is that right?

Answer 6

Good..

Answer 7

Then you do the differentiation of .

Answer 8

\[\frac{d}{dx}(\sin(\color{red}{x})) = \cos(x) \times \frac{d}{dx} (\color{red}x)\]

Answer 9

\[\frac{d}{dx} (x) = 1\\ so, \; \; \frac{d}{dx}(\sin(x))= \cos(x) \times 1 = \cos(x)\]

Answer 10

yes

Answer 11

Similarly do that..

Answer 12

\[\frac{d}{dx} (\sin(xy)) = \cos(xy) \times \frac{d}{dx}(xy)\]

Answer 13

oooh now i get it, so will it ultimately be cos x+ cos(xy)d/dx

Answer 14

Now, here product rule applies..

Answer 15

wait..

Answer 16

ok

Answer 17

Final answer?

Answer 18

\[d(\sin(xy)) = \cos(xy)[x \frac{dy}{dx} + y]\]

Answer 19

Is this okay?