If sin(x+y)=3x−2y, then dydx=
Correct. The chain rule is the basis for implicit differentiation as well as the differentiation of a composite function. cos(x+y)(1+dydx)=3−2dydxcos(x+y)(1+dydx)=3−2dydx cos(x+y)+cos(x+y)dydx=3−2dydxcos(x+y)+cos(x+y)dydx=3−2dydx (2+cos(x+y))dydx=3−cos(x+y)(2+cos(x+y))dydx=3−cos(x+y) dydx=3−cos(x+y)2+cos(x+y)
To find dy/dx, we need to take the partial derivative of the given equation with respect to x and y:
∂/∂x(sin(x+y)) = 3
∂/∂y(sin(x+y)) = -2
Using the chain rule, we can now find the derivative of y with respect to x:
dy/dx = (∂/∂x(sin(x+y))) / (∂/∂y(sin(x+y)))
dy/dx = 3 / (-2)
dy/dx = -3/2
Therefore, the derivative of y with respect to x is -3/2.
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