Question

Finding Derivatives Implicity In Exercise,Find dy/dx implicity.
x2y - ey - 4 = 0

Answer 1

`(dy)/(dx) = (2xy)/(e^y-x^2)`

Explanation:

We are given the following in the question:

`x^2y - e^y - 4 = 0`

We have to find the derivative of the given expression implicitly.

Formula:

`\text{Product rule:}\\\\(d(a.b))/(dx) = b(da)/(dx) + a(db)/(dx)\\\\(d(x^n))/(dx) = nx^(n-1)\\\\\frac{d(\text{Constant})}{dx} = 0`

The derivation takes place in the following manner:

`x^2y - e^y - 4 = 0\\\text{Differentiating both sides we get,}\\(x^2dy + 2xdxy)-e^ydy = 0\\(x^2-e^y)dy + (2xy)dx = 0\\(x^2-e^y)dy = - (2xy)dx\\\\(dy)/(dx) = (-2xy)/(x^2-e^y)`

Thus, the implicit differential is given by:

`(dy)/(dx) = (2xy)/(e^y-x^2)`