5ymmetrica1 said:Homework Statement
For the curve x2+3xy+y2 = 5
show that [itex]\frac{dy}{dx} =- \frac{2x+3y}{3x+2y}[/itex]
Homework Equations
N.A.
The Attempt at a Solution
2x + 3xy + 2y[itex]\frac{dy}{dx}[/itex] = 0
3x + 2y [itex]\frac{dy}{dx}[/itex] = -2x+ 3y
∴ [itex]\frac{dy}{dx} =- \frac{2x+3y}{3x+2y}[/itex]
have I done this correctly?
Implicit differentiation is a technique used in calculus to find the derivative of a function that is defined implicitly, rather than explicitly. This means that the function is not written in the form of y = f(x), but rather as an equation where both x and y are present.
Implicit differentiation is used when it is difficult or impossible to solve for y in terms of x in an equation. This often occurs with curves or shapes that cannot be represented by a simple function.
Explicit differentiation is used to find the derivative of a function that is defined explicitly in terms of x. This means that the function is written as y = f(x). Implicit differentiation, on the other hand, is used to find the derivative of a function that is defined implicitly, where both x and y are present in the equation.
The process of implicit differentiation involves taking the derivative of both sides of an equation with respect to x. The chain rule is then used to differentiate any terms that contain y, and the derivative of x is simply 1. The resulting equation can then be solved for y' to find the derivative of the original function.
Implicit differentiation is used in a variety of fields, including physics, engineering, and economics. It is particularly useful in finding the slopes of curves and surfaces, as well as in optimization problems where the derivative is used to find maximum or minimum values of a function.