Glossary
Chain Rule
A fundamental differentiation rule used to find the derivative of composite functions. It states that the derivative of a composite function is the derivative of the outer function (evaluated at the inner function) multiplied by the derivative of the inner function.
Example:
To find the derivative of , you apply the Chain Rule to get .
Composite Functions
Functions formed by combining two or more functions, where the output of one function becomes the input of another. They are typically written as $f(g(x))$.
Example:
If and , then is a composite function.
Function Notation
A common notation for derivatives using primes, such as $f'(x)$ or $\frac{d}{dx}(f(x))$. For the Chain Rule, it is expressed as $\frac{d}{dx}(f(g(x))) = f'(g(x)) \cdot g'(x)$.
Example:
If , using Function Notation, .
Inner Function
In a composite function $f(g(x))$, the *inner function* is $g(x)$, which is the function applied first to the independent variable.
Example:
For the function , the inner function is .
Leibniz Notation
A common notation for derivatives, often expressed as a ratio of differentials like $\frac{dy}{dx}$. For the Chain Rule, it is written as $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$.
Example:
When differentiating , you might use Leibniz Notation by setting and then finding and to combine them.
Outer Function
In a composite function $f(g(x))$, the *outer function* is $f(x)$, which operates on the output of the inner function.
Example:
For the function , the outer function is (where ).