A curve has the equation: x^4 + 2x -xy - y^3 - 10=0. Find dy/dx in terms of x and y.

We need to differentiate all values with respect to x. Therefore for the first two terms, multiply by the power and then subtract 1 from the original power. Therefore 4(x^4-1) + (2)(2x^2-1) which gives 4x³ + 4x for the first two terms.
For the 3rd term, it contains an x and y value. Differentiating x gives (1)(y) = y
Implicit differentiation is required for the y value, in which the rule d/dx (f(y)) = d/dy (f(y))dy/dx gives (x)(1)(d/dx) = xdy/dx 3. Again the 4th term requires to differentiated implicitly. This gives 3y²(dy/dx).
Therefore we end up with 4x³ + 4x - y -x(dy/dx) - 3y²(dy/dx) = 0 4. Finally, take all dy/dx terms to one side and take out dy/dx as a common term: 4x^3 + 4x -y = dy/dx (3y^2 +x)
Therefore dy/dx = (4x³ + 4x -y)/(3y² +x)