Derivatives

Overview

Let $f$ be a function defined on some open interval $(a,b)$. The derivative of $f$, denoted $f^{\prime }(x)$ or $Df$, is given by

provided this limit exists. The fraction we are taking the limit on is known as the difference quotient.

Differentials

Leibniz preferred the following notation for defining a derivative:

$\mathrm{\Delta }$ is called the difference operator, $dy$ and $dx$ are called differentials, and the derivative $dy/dx$ is called a differential quotient.

The differential quotient was the quotient of infinitesimal quantities Leibniz imagined as entirely new numbers greater than zero but smaller than every positive real number.

Identities

The following identities assume $f$ and $g$ are functions defined on a common interval. Assume their derivatives are defined at point $x$. Then:

Addition

Subtraction

Multiplication

Also known as the product rule. $$(f \cdot g)'(x) = f'(x)g(x) + f(x)g'(x)$$

Division

Chain Rule

Suppose $v$ is differentiable at $x$ and $u$ is differentiable at $v(x)$. Then $f=u\circ v$ is differentiable at $x$ and

Implicit Differentiation

Implicit differentiation refers to the differentiation of an implicit function. They leverage the chain rule by maintaining that the value of the function is defined in terms of the others as arguments.

Logarithmic Differentiation

Let $f(x)$ be a differentiable function and define $g(x)=\ln |f(x)|$. Then, by virtue of the chain rule,

Continuity

Let $f$ be a function differentiable at $x$. Then $f$ is continuous at $x$.

We say a function $f$ is continuously differentiable if $f$ is differentiable and its derivative is continuous.

Inverse Function Rule

Let $f$ be a differentiable function on $[a,b]$ with inverse $g$. If $f^{\prime }(x)\ne 0$ exists for some $x\in (a,b)$, then $g^{\prime }(y)\ne 0$ also exists at corresponding point $y=f(x)$. Moreover,

Extremums

Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. A critical point $c$ of $f$ is a point $c\in (a,b)$ where $f^{\prime }(c)=0$. Then the only places where extrema can occur are:

1. At the endpoints;
2. At the critical points of $f$.

Conversely, if $f$ has a relative minimum or relative maximum at $c\in (a,b)$, then $f^{\prime }(c)=0$.

Rolle's Theorem

Let $f$ be a function continuous everywhere on closed interval $[a,b]$ and differentiable on open interval $(a,b)$. If $f(a)=f(b)$, then there exists a $c\in (a,b)$ such that $f^{\prime }(c)=0$.

Mean Value Theorem

Let $f$ be a function continuous everywhere on closed interval $[a,b]$ and differentiable on open interval $(a,b)$. Then there exists a $c\in (a,b)$ such that

Cauchy's Mean Value Theorem

Let $f$ and $g$ be two functions continuous on a closed interval $[a,b]$ and having derivatives in the open interval $(a,b)$. Then, for some $c\in (a,b)$, we have

Signedness Properties

Let $f$ be a function continuous on closed interval $[a,b]$ and differentiable on $(a,b)$. Then:

• If $f^{\prime }(x)>0$ for all $x\in (a,b)$, $f$ is strictly increasing on $[a,b]$.
• If $f^{\prime }(x)<0$ for all $x\in (a,b)$, $f$ is strictly decreasing on $[a,b]$.
• If $f^{\prime }(x)=0$ for all $x$ in $(a,b)$, $f$ is constant throughout $[a,b]$.

First-Derivative Test

Let $f$ be continuous on $[a,b]$. Assume $f$ is differentiable on $(a,b)$ except possibly at $c\in (a,b)$. Then:

• If $f^{\prime }(x)>0$ for all $x<c$ and $f^{\prime }(x)<0$ for all $x>c$, then $f$ has a relative maximum at $c$.
• If $f^{\prime }(x)<0$ for all $x<c$ and $f^{\prime }(x)>0$ for all $x>c$, then $f$ has a relative minimum at $c$.

Second-Derivative Test

Let $c$ be a critical point of $f$ in $(a,b)$. Also assume $f$ is twice differentiable on $(a,b)$. Then:

• If $f^{″}(x)<0$ for $x\in (a,b)$, $f$ has a relative maximum at $c$.
• If $f^{″}(x)>0$ for $x\in (a,b)$, $f$ has a relative minimum at $c$.