Final answer:
To find dy/dx by implicit differentiation, differentiate both sides of the equation with respect to x. The derivative of sin(xy^2) with respect to x is cos(xy^2) * (y^2 * dx/dx + x * d(y^2)/dx). Simplify the expression to find dy/dx.
Step-by-step explanation:
To find dy/dx by implicit differentiation, we will differentiate both sides of the equation with respect to x.
We have the equation 5+4x=sin(xy^2). Taking the derivative of both sides, the left side will be 0 since it is a constant. On the right side, we use the chain rule.
The derivative of sin(xy^2) with respect to x can be written as cos(xy^2) * (y^2 * dx/dx + x * d(y^2)/dx).
Simplifying further, dy/dx = (cos(xy^2) * (2xy^2 + x * 2y * dy/dx)) / (1 - x * 2y^2 * cos(xy^2)).