Introduction
Precalculus
Getting Started
Calculus 1
1. Limits
3. Derivatives
5. Applications of Derivatives
6. 1. Increasing and Decreasing Function
  2. Minimum and Maximum Values
7. Integrals
8. 1. Indefinite Integrals
  2. Computing Indefinite Integrals
  3. Substitution Rule for Indefinite Integrals
  4. Area Problem
Calculus 2
1. Integration Techniques
2. 1. Integration by Parts
  2. Integrals Involving Trig Functions
  3. Trig Substitutions
  4. Partial Fractions
  5. Integrals Involving Roots
  6. Integrals Involving Quadratics
3. Parametric Equations and Polar Coordinates
4. 1. Differentiation Formulas
  2. Product and Quotient Rule
  3. Chain Rule
  4. Implicit Differentiation
  5. Higher Order Derivatives
  6. Logarithmic Differentiation
5. Series and Sequences
6. 1. Indefinite Integrals
  2. Computing Indefinite Integrals
  3. Substitution Rule for Indefinite Integrals
  4. Area Problem
7. Vectors
8. 3-Dimensional Space
Calculus 3
1. LimitThree Dimensional Coordinate Systems
2. 1. Equations of Lines
  2. Equations of Planes
  3. Quadratic Surfaces
  4. Functions of Multiple Variables
3. Partial Derivatives
4. 1. Higher Order Partial Derivatives
  2. Differentials
  3. Chain Rule
  4. DirectioChain Rulenal Derivatives
  5. Gradient
5. Multiple Integrals
6. 1. Iterated Integrals in Cylindrical Coordinates
  2. Double Integrals
  3. Double Integrals in Polar Coordinates
  4. Triple Integrals in Spherical Coordinates
  5. Change of Variables
  6. Surface Area
7. Line Integrals
8. 1. Line Integrals With Respect to Arc Length
  2. Double Integrals in Polar Coordinates
  3. Line Integrals With Respect to x and y
  4. Line Integrals of Vector Fields
  5. Fundamental Theorem of Line Integrals
  6. Conservative Vector Fields
  7. Potential Functions
  8. Green's Theorem
9. Surface Integrals
10. 1. Parametric Surfaces
  2. Surface Integrals
  3. Surface Integrals of Vector Fields
  4. Stokes' Theorem
  5. Divergence Theorem
Cheatsheet

Implicit differentiation

What is Implicit differentiation?

Implicit differentiation is a technique that allows us to differentiate equations that cannot be expressed in the form $y = f(x)$. It is based on the fact that the derivative of a function is equal to the derivative of the dependent variable with respect to the independent variable, multiplied by the derivative of the independent variable with respect to itself.

Here is an example of implicit differentiation using KaTeX syntax:

Let's say we want to differentiate the equation $x^2 + y^2 = 9$. This equation cannot be written in the form $y = f(x)$, so we cannot use the standard rules of calculus to differentiate it. Instead, we can use implicit differentiation to find the derivative of $y$ with respect to $x$.

We start by differentiating both sides of the equation with respect to $x$:

$$\frac{d}{dx} (x^2 + y^2) = \frac{d}{dx} 9$$

Using the chain rule, we can write the derivative of the left-hand side as:

$$2x + 2y \cdot \frac{dy}{dx} = 0$$

We can then solve for $\frac{dy}{dx}$:

$$\frac{dy}{dx} = -\frac{2x}{2y} = -\frac{x}{y}$$

So, the derivative of $y$ with respect to $x$ is $-\frac{x}{y}$.

This is an example of how implicit differentiation can be used to find the derivative of a function that is not explicitly defined in terms of $x$ and $y$. It is a useful technique when we are given an equation that cannot be written in the form $y = f(x)$ and we want to find the derivative of one of the variables with respect to the other.