Glossary
Chain rule
A fundamental rule in calculus used to differentiate composite functions, where one function is nested inside another.
Example:
When differentiating a term like y³ with respect to x in an implicit equation, you apply the chain rule to get 3y² multiplied by dy/dx.
Explicitly
An equation is written explicitly when one variable (e.g., y) is expressed solely in terms of the other variable (e.g., x).
Example:
The equation y = 5x² - 3 is written explicitly, making it straightforward to determine the value of y for any given x.
Implicit differentiation
A technique used to find the derivative of implicit functions by differentiating each term with respect to a chosen variable (usually x) and applying the chain rule for terms involving the other variable (y).
Example:
To find the slope of the tangent line to the curve defined by x³ + y³ = 1 at a specific point, you would use implicit differentiation to determine dy/dx.
Implicit functions
Equations involving both x and y where y cannot easily be expressed solely in terms of x, or vice versa, but still define a relationship between the variables.
Example:
The equation x² + y² = 25, representing a circle, is an implicit function because y is not isolated on one side.
Inverse functions
Functions that reverse the effect of another function, meaning if f(a) = b, then the inverse function f⁻¹(b) = a.
Example:
The derivative of inverse functions like arcsin(x) or arctan(x) can be elegantly derived using the principles of implicit differentiation.
Product rule
A rule used to find the derivative of a function that is the product of two or more differentiable functions.
Example:
If an implicit equation contains a term like 6xy, you must use the product rule to differentiate it with respect to x, treating 6x and y as separate functions.
Quotient rule
A rule used to find the derivative of a function that is the ratio (quotient) of two differentiable functions.
Example:
While not explicitly shown in the notes' examples, if an implicit function involved a term like (x²)/(y), finding its derivative would require applying the quotient rule.