find the implicit function of x^3+3^y=3xy
Not sure if you can solve it explicitly. A 3-d plot is possible with most graphics software. The solution curve is given by the contour x^3+3^y-3xy=0.
See for example:
http://imageshack.us/photo/my-images/163/1313500261.png/
examine this closely.
3x^2 dx+3^y * ln3*dy=3y dx + 3x dy
dy/dx= 3(y-x^2) / (ln3*3^y -3x)
check my algebra, I gathered terms in my head.
To find the implicit function of the equation x^3 + 3^y = 3xy, we need to differentiate the equation with respect to x while treating y as a function of x using the chain rule.
Let's differentiate both sides of the equation with respect to x:
d/dx (x^3 + 3^y) = d/dx (3xy)
The derivative of x^3 with respect to x is 3x^2.
The derivative of 3^y with respect to x involves the chain rule. We have:
d/dx (3^y) = d/dy (3^y) * dy/dx
To find d/dy (3^y), we can use the chain rule for exponentials. The derivative of 3^y with respect to y is (ln 3) * (3^y).
Now, let's find dy/dx by differentiating the term 3xy:
dy/dx = (d/dx)(3xy) = 3y + 3x * (dy/dx)
Now, we can substitute these derivatives back into the equation:
3x^2 + (ln 3) * (3^y) * (dy/dx) = 3y + 3x * (dy/dx)
Now, we can isolate the terms involving dy/dx:
(dy/dx) * [(ln 3) * (3^y) - 3x] = 3y - 3x^2
Finally, we can solve for dy/dx:
dy/dx = (3y - 3x^2) / [(ln 3) * (3^y) - 3x]
This is the implicit function for the given equation x^3 + 3^y = 3xy.