Question

Find
dy/dx by implicit differentiation.
y = sin xy

Answer 1

Use the product rule for sin xy.

Answer 2

okay @shamil98 is here, he's way better explaining things

Answer 3

o-o

Answer 4

dy/dx (y) = dy/dx (sin xy)
y' = cosxy + cosxy'
-cosxy = y'(cos x - 1)
-cosxy/(cos x- 1) = y'

Answer 5

thank you @shamil98 :)

Answer 6

Let's start over:
Find
dy/dx by implicit differentiation.
y = sin xy

We do not use the product rule on sin xy; rather, we apply it to xy only.

Taking the derivative of both sides,

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

Answer 7

This results in \[\frac{ dy }{ dx }=\cos(xy)*\frac{ d }{dx }(xy)\]

Answer 8

... which simplifies to\[\frac{ dy }{ dx }=\cos (xy) * [x*\frac{ dy }{ dx }+y \frac{ dx }{ dx }].\]

Placing both dy/dx terms on the left side, we get\[\frac{ dy }{ dx }-\cos(xy)*x*\frac{ dy }{ dx }=\cos(xy)*y.\]
Now factor dy/dx out of the two terms on the left.

Finally, solve for dy/dx.

Answer 9

kind of late for that :/ but k