Find dy/dx by implicit differentiation. e^(y)*cos(x) = 8 + sin(xy)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The derivative `dy/dx` has to be determined for the expression `e^(y)*cos(x) = 8 + sin(xy)`

Using implicit differentiation

`e^y*(dy/dx)*cos x + e^y*(-sin x) = 0 + cos(x*y)*(x*(dy/dx)+y)`

=> `(dy/dx)(e^y*cos x - x*cos(xy)) = e^y*sin x + cos(x*y)*y`

=> `dy/dx = (e^y*sin x + cos(x*y)*y)/(e^y*cos x - x*cos(xy))`

The derivative `dy/dx` for `e^(y)*cos(x) = 8 + sin(xy)` is `dy/dx = (e^y*sin x + cos(x*y)*y)/(e^y*cos x - x*cos(xy))`

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