5.2 Implicit differentiation using the chain rule

Implicit differentiation extends the chain rule to equations where you can't easily isolate one variable. Many important curves and surfaces in multivariable calculus are defined by equations like $F(x, y) = 0$ rather than explicit formulas $y = f(x)$. This technique lets you compute derivatives, tangent lines, and tangent planes for all of them.

Implicit Differentiation

Implicit Functions and Differentiation

An implicit function defines a relationship between variables without solving for one in terms of the others. The equation $x^2 + y^2 = 1$ defines a circle, but you can't write the whole circle as a single function $y = f(x)$. Instead, the relationship $F(x, y) = 0$ defines $y$ implicitly.

To differentiate an implicit equation, you treat $y$ as a function of $x$ and apply the chain rule wherever $y$ appears. Here's the process:

1. Start with your equation (e.g., $x^2 + y^2 = 1$).
2. Differentiate both sides with respect to $x$. Every time you differentiate a term involving $y$, multiply by $\frac{dy}{dx}$ (that's the chain rule doing its work).
3. Collect all terms containing $\frac{dy}{dx}$ on one side.
4. Solve algebraically for $\frac{dy}{dx}$.

For the circle example: differentiating gives $2x + 2y\frac{dy}{dx} = 0$, so $\frac{dy}{dx} = -\frac{x}{y}$.

The total differential connects to this idea. For a function $z = f(x, y)$, the total differential is:

$dz = \frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$

This expression captures the total change in $f$ due to small changes in both $x$ and $y$. It's useful for approximating changes in a function and for analyzing error propagation in applied problems.

Implicit Function Theorem

The Implicit Function Theorem tells you when an equation $F(x, y) = 0$ actually defines $y$ as a smooth function of $x$ near a given point. The conditions are:

• $F(a, b) = 0$ (the point lies on the curve), and
• $\frac{\partial F}{\partial y}(a, b) \neq 0$ (the curve isn't "vertical" at that point).

When both conditions hold, there exists a unique function $y = f(x)$ defined near $x = a$ satisfying $F(x, f(x)) = 0$, and its derivative is:

$\frac{dy}{dx} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$

This formula is worth memorizing. It gives you the derivative directly from the partial derivatives of $F$, without doing the full implicit differentiation procedure each time. Notice why $\frac{\partial F}{\partial y} \neq 0$ matters: it's in the denominator.

For the circle $F(x,y) = x^2 + y^2 - 1$, you get $\frac{dy}{dx} = -\frac{2x}{2y} = -\frac{x}{y}$, matching the result from before.

The theorem generalizes to higher dimensions as well, guaranteeing when implicit equations define surfaces or higher-dimensional objects locally.

Curves and Surfaces

Level Curves and Parametric Equations

Level curves (contour lines) are the sets of points where a function takes a constant value. For $f(x, y)$, the level curve at value $c$ is the set of all $(x, y)$ satisfying $f(x, y) = c$. These are exactly the implicit equations you've been differentiating. Topographic maps use this idea: each contour line represents constant elevation.

Level curves give you a way to visualize a function of two variables in the plane. Where level curves are packed close together, the function is changing rapidly; where they're spread apart, it's changing slowly.

Parametric equations describe curves using an independent parameter. A curve in the plane can be written as $x = x(t)$, $y = y(t)$, where $t$ is the parameter. This representation is more flexible than implicit or explicit forms because it can handle self-intersections, loops, and curves that fail the vertical line test. It also lets you compute velocity and acceleration along the curve directly.

Tangent Planes

A tangent plane is the multivariable analogue of a tangent line. For a surface $z = f(x, y)$, the tangent plane at the point $(a, b, f(a, b))$ is:

$z - f(a, b) = \frac{\partial f}{\partial x}(a, b)(x - a) + \frac{\partial f}{\partial y}(a, b)(y - b)$

This is a linear approximation of the surface near that point, built from the two partial derivatives.

The normal vector to the tangent plane points perpendicular to the surface and is given by:

$\vec{n} = \left(\frac{\partial f}{\partial x}(a, b),\ \frac{\partial f}{\partial y}(a, b),\ -1\right)$

For a surface defined implicitly by $F(x, y, z) = 0$, the gradient $\nabla F = \left(\frac{\partial F}{\partial x}, \frac{\partial F}{\partial y}, \frac{\partial F}{\partial z}\right)$ serves as the normal vector, and the tangent plane equation becomes:

$\frac{\partial F}{\partial x}(x - a) + \frac{\partial F}{\partial y}(y - b) + \frac{\partial F}{\partial z}(z - c) = 0$

Tangent planes are used for local linear approximation of surfaces and show up in optimization problems, such as finding the closest point on a surface to a given point in space.