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EleoNora [17]
3 years ago
14

Sin(xy)-x=0 find dy/Dx

Mathematics
1 answer:
MariettaO [177]3 years ago
4 0
If you're using the app, try seeing this answer through your browser:  brainly.com/question/2989024

——————————

You have  y  as an implicit function of  x:

sin(xy) – x = 0


Use implicit differentiation. As  y  is a function of  x, then you must apply the chain rule there:

  d                                 d
—— [ sin(xy) – x ]  =  —— (0)
 dx                               dx


  d                           d               d
—— [ sin(xy) ]  –  —— (x)  =  —— (0)
 dx                         dx             dx

  d
—— [ sin(xy) ]  –  1  =  0
 dx

  d
—— [ sin(xy) ]  =  1
 dx
 
                   d
cos(xy)  ·  —— (xy) = 1
                  dx


Now, apply the product rule for that last derivative:

\mathsf{cos(xy)\cdot \left[\dfrac{d}{dx}(x)\cdot y+x\cdot \dfrac{dy}{dx}\right]=1}\\\\\\
\mathsf{cos(xy)\cdot \left[1\cdot y+x\cdot \dfrac{dy}{dx}\right]=1}\\\\\\
\mathsf{y\,cos(xy)+x\,cos(xy)\cdot \dfrac{dy}{dx}=1}


              dy
Isolate  —— :
              dx

                   dy
x cos(xy) · ——  =  1 – y cos(xy)
                   dx


Assuming  x cos(xy) ≠ 0,

 dy          1 – y cos(xy)
——  =  ————————    <———    this is the answer.
 dx            x cos (xy)


I hope this helps. =)

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