Differential calculus

Differential calculus is one of the fundamental branches of mathematics, providing the theoretical foundation for understanding rates of change and optimization problems. The central concept of differential calculus is the derivative, which measures the instantaneous rate of change of a function. This concept allows us to analyze motion, optimize functions, understand the behavior of curves, and solve complex problems involving changing quantities.

Definition of the derivative

Consider a function $y=f(x)$, where we are interested in the instantaneous slope at the point $(a,f(a))$. We can first consider the secant line that connects the point $(a,f(a))$ to another point on the function $(a+\Delta x,f(a)+\Delta y)$, as shown in the graph below:

@@@ Figure [differential-1]: Secant line between two points on a function $y=f(x)$ @@@

By taking the limit as $\Delta x$ goes to zero, the slope of the secant line approaches the instantaneous rate of change of the function $f(x)$ at $x=a$. The derivative of a function represents the instantaneous rate of change of that function with respect to its independent variable. In geometric terms, the derivative at a point gives the slope of the tangent line to the curve at that point.

The derivative of a function $f(x)$ at a point $x$ is defined as the limit:

$$f^{\prime }(x)=\underset{\Delta x\to 0}{\lim }\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

This limit, when it exists, gives us the instantaneous rate of change of $f(x)$ at the point $x$.

Alternative notations

The derivative can be expressed using several equivalent notations:

• $f^{\prime }(x)$ (Lagrange notation)
• $\frac{df}{dx}$ (Leibniz notation)
• $D_{x}f$ (operator notation)
• $\dot{f}$ (Newton notation, often used for time derivatives)

Geometric interpretation

The derivative $f^{\prime }(x)$ represents the slope of the tangent line to the curve $y=f(x)$ at the point $(x,f(x))$. This geometric interpretation connects the algebraic concept of limits with the visual understanding of curves and their behavior.

Physical interpretation

In physics, if $s(t)$ represents position as a function of time, then:

• $s^{\prime }(t)=v(t)$ represents velocity (rate of change of position)
• $s^{\prime \prime }(t)=a(t)$ represents acceleration (rate of change of velocity)

The second derivative

The second derivative, denoted as $f^{\prime \prime }(x)$ or $\frac{d^{2}f}{d{x}^2}$, is the derivative of the first derivative. It provides information about the concavity and rate of change of the slope of the original function.

Geometric interpretation

• If $f^{\prime \prime }(x)>0$, the function is concave up (curves upward).
• If $f^{\prime \prime }(x)<0$, the function is concave down (curves downward).
• If $f^{\prime \prime }(x)=0$, there may be a point of inflection.

Physical interpretation

In physics, if $s(t)$ represents position:

• $s^{\prime }(t)=v(t)$ is velocity
• $s^{\prime \prime }(t)=a(t)$ is acceleration

Derivatives of common functions

Understanding the derivatives of basic functions is essential for applying differential calculus to complex problems. The following table summarizes the derivatives of the most commonly encountered functions:

Function	Derivative
$c$ (constant)	$0$
$x$	$1$
$x^n$	$n{x}^{n-1}$
$e^x$	$e^x$
$a^x$	$a^x\ln a$
$\ln x$	$\frac{1}{x}$
${\log }_{a}x$	$\frac{1}{x\ln a}$
$\sin x$	$\cos x$
$\cos x$	$-\sin x$
$\tan x$	${\sec }^{2}x$
$\cot x$	$-{\csc }^{2}x$
$\sec x$	$\sec x\tan x$
$\csc x$	$-\csc x\cot x$
${\sin }^{-1}x$	$\frac{1}{\sqrt{1-x^2}}$
${\cos }^{-1}x$	$-\frac{1}{\sqrt{1-x^2}}$
${\tan }^{-1}x$	$\frac{1}{1+x^2}$

Properties of derivatives

Linearity of differentiation

Differentiation is a linear operation, which means it satisfies two important properties:

Constant multiple rule:

$$\frac{d}{dx}[cf(x)]=c\frac{df}{dx}$$

where $c$ is any constant.

Sum and difference rule:

$$\frac{d}{dx}[f(x)\pm g(x)]=\frac{df}{dx}\pm \frac{dg}{dx}$$

These properties allow us to differentiate complex expressions by breaking them into simpler components.

The product rule

When differentiating the product of two functions, we cannot simply multiply their derivatives. Instead, we use the product rule:

$$\frac{d}{dx}[f(x)g(x)]=f(x)\frac{dg}{dx}+g(x)\frac{df}{dx}$$

The quotient rule

For the quotient of two functions, we use the quotient rule:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)\frac{df}{dx}-f(x)\frac{dg}{dx}}{[g(x){]}^2}$$

This rule is often remembered by the mnemonic “low d-high minus high d-low, over low squared.”

The chain rule

The chain rule is used to differentiate composite functions. If $y=f(g(x))$, then:

$$\begin{array}{rl}\frac{dy}{dx}& =\frac{df}{dg}\cdot \frac{dg}{dx}\\ & =f^{\prime }(g(x))\cdot g^{\prime }(x)\end{array}$$

In Leibniz notation, where $y=f(u)$ and $u=g(x)$, then the chain rule is expressed as:

$$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$$

For more complex compositions, the chain rule extends naturally:

$$\frac{d}{dx}[f(g(h(x)))]=f^{\prime }(g(h(x)))\cdot g^{\prime }(h(x))\cdot h^{\prime }(x)$$

L’Hôpital’s rule

L’Hôpital’s rule provides a method for evaluating limits of indeterminate forms, particularly those of the type $\frac{0}{0}$ or $\frac{\infty }{\infty }$.

L’Hôpital’s rule

If ${\lim }_{x\to a}f(x)=0$ and ${\lim }_{x\to a}g(x)=0$ (or both limits are $\pm \infty$), and if ${\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$ exists, then:

$$\underset{x\to a}{\lim }\frac{f(x)}{g(x)}=\underset{x\to a}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$$

L’Hôpital’s rule can be applied repeatedly if the resulting limit is still indeterminate.

Critical points and optimization

Critical points are locations where the derivative of a function is zero or undefined. These points are essential for understanding the behavior of functions and solving optimization problems.

Definitions

Critical point

A point $x=c$ is a critical point of $f(x)$ if $f^{\prime }(c)=0$ or $f^{\prime }(c)$ is undefined.

Classification of critical points

To determine the nature of a critical point, we use the following tests:

First derivative test:

• If $f^{\prime }(x)$ changes from positive to negative at $x=c$, then $f$ has a local maximum at $c$.
• If $f^{\prime }(x)$ changes from negative to positive at $x=c$, then $f$ has a local minimum at $c$.
• If $f^{\prime }(x)$ does not change sign at $x=c$, then $f$ has neither a maximum nor minimum at $c$.

Second derivative test:

If $f^{\prime }(c)=0$, then:

• If $f^{\prime \prime }(c)>0$, then $f$ has a local minimum at $x=c$.
• If $f^{\prime \prime }(c)<0$, then $f$ has a local maximum at $x=c$.
• If $f^{\prime \prime }(c)=0$, the test is inconclusive.

Points of inflection

A point of inflection occurs where the concavity of the function changes. This happens when:

• $f^{\prime \prime }(c)=0$ and $f^{\prime \prime }(x)$ changes sign at $x=c$.

Curvature and radius of curvature

Curvature measures how sharply a curve bends at a given point. It is closely related to the second derivative but provides a more geometric understanding of curve behavior.

Definition of curvature

For a curve $y=f(x)$, the curvature $\kappa$ (kappa) at a point is defined as:

$$\kappa =\frac{|f^{\prime \prime }(x)|}{[1+(f^{\prime }(x){)}^2{]}^{3/2}}$$

Radius of curvature

The radius of curvature $R$ is the reciprocal of the curvature:

$$R=\frac{1}{\kappa }=\frac{[1+(f^{\prime }(x){)}^2{]}^{3/2}}{|f^{\prime \prime }(x)|}$$

The radius of curvature represents the radius of the circle that best approximates the curve at a given point. This circle is called the osculating circle or circle of curvature. Applications include:

• Engineering: designing smooth transitions in roads and railways.
• Physics: understanding the motion of particles along curved paths.
• Computer graphics: creating smooth curves and surfaces.