Question

A curve is given by x^(2)y+xe^(y)=sinx Use implicit differentiation to find (dy)/(dx).

Answer 1

To find (dy)/(dx), we differentiate both sides of the equation using implicit differentiation. After differentiating each term, we solve for (dy)/(dx) by simplifying the equation.

Step-by-step explanation:

To find (dy)/(dx) using implicit differentiation, we differentiate both sides of the equation with respect to x. Let's start by differentiating x^(2)y using the product rule:

(d/dx)(x^(2)y) = 2xy + x^(2)(dy/dx)

Next, differentiate xe^(y) using the chain rule:

(d/dx)(xe^(y)) = e^(y) + xe^(y)(dy/dx)

Now, differentiate sinx which is a function of x: (d/dx)(sinx) = cosx

Substitute these results back into the equation: 2xy + x^(2)(dy/dx) + e^(y) + xe^(y)(dy/dx) = cosx

Now, solve for (dy/dx):

x^(2)(dy/dx) + xe^(y)(dy/dx) = cosx - 2xy - e^(y)

dy/dx = (cosx - 2xy - e^(y))/(x^(2) + xe^(y))