Implicit Function

An implicit function includes the dependent variable within an equation, like x2+y2=1, without solving for it. Instead of giving y explicitly as y = f(x). Implicit differentiation lets you find dy/dx by differentiating both sides using the chain rule without solving for y first. Implicit differentiation is useful when it’s hard or impossible to solve for y directly.
 

Cannot be written as y = f(x)

The dependent variable y isn't isolated; both x and y appear together in the same equation

Always in the form f(x, y) = 0

The equation combines x and y like x2+y2-1 = 0 for a circle

Frequently non-linear and multivariable

Many implicit functions are polynomials or complex relations, including many variables

Combines dependent and independent terms.

These cannot be easily solved for y because it stays within the equation with x.

It may not pass the vertical line test.

An implicit relation does not always pass the vertical line test because:
A vertical line might cross its graph more than once.
That means y is not a single-valued function of x.
 

While solving an equation involving both x and y together f(x, y) = 0, you can't isolate y very easily, you directly differentiate both sides without solving a variable first. With respect to x treating y as a function of x and use the chain rule whenever you differentiate a term including y.

Differentiate each term in the equation with respect to x, adding dydx while differentiating anything with y
Collect all dydx, terms on one side
Solve for dydx, the result will usually involve both x and y

x2+xy+y=0

Differentiate both sides w.r.t.x:

ddx(x2)+ddx(xy)+ddx(y)=0

Becomes

2x+(xdydx+y1)+dydx=0

Combine dydx terms and isolate

(x+1)dydx+(2x+y)=0dydx=-2x+yx+1

You get dydx directly without needing to solve for y first.
 

The implicit function theorem helps to turn complicated relationships into simpler functions of real variables. It shows that, even if a relation doesn’t define a function globally, it can behave like a function locally. For example, given an equation like F(x, y)=0, the theorem ensures that we can solve for y as a smooth function of x. Certain partial derivatives are non-zero, indicating how the function changes as x changes.
 

Real life application is important in many fields such as economics, physics, chemistry, The uses are explained below;

• Economics-marginal rate of substitution: Economists use implicit differentiation to analyze trade-offs between goods to find how much more of one good is needed to remain equally satisfied when consuming less of another, for example for utility equation U(x, y)=k, implicit differentiate gives dy/dx = -(Ux/Uy)
• Ladder sliding problem: This is used by teachers and students to solve motion problems in physics. As a ladder slides down a wall, its top moves downward while the base slides outward. To relate the rates of change of these positions, we use implicit differentiation. For example, if the ladder has a fixed length h, and its position forms a right triangle, the relationship is: x2+y2=h2 
• By differentiating both sides with respect to time t, we can find dydt (how fast the top is falling) in terms of dxdt (how fast the base is sliding).
• Physics-motion with constraints: Physicists analyzing moving or constrained systems often deal with motion equations in implicit form. Implicit differentiation is used to find quantities ;like velocity and acceleration when dependent on multiple variables. For example, pendulum constraints x2+y2=l2; differentiate implicitly to find angular acceleration
• Fluid dynamics/ engineering: This is used by engineers modeling pressure and flows, where equations relating to pressure, velocity, and geometry are often implicit. Implicit differentiation helps to find the gradients, such as in the Navier-stokes equation, to predict the fluid behavior.
• Thermodynamics/ chemistry: Implicit differentiation is used in thermodynamics. This helps in finding how the pressure changes with volume or temperature in gas laws like the Van der Waals equation, where variables are interdependent.