Derivative of a Function

Definition of the Derivative

The derivative of a function f(x) at a point x is defined as the limit of the difference quotient: $$ \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}$$ If we introduce the notation \( h=\Delta x \), the same limit can also be written in the equivalent form: $$ \lim_{h \to 0} \frac{f(x+h )-f(x)}{h}$$ Both expressions represent the same concept. The only difference is the symbol used for the increment of the variable.

In practical terms, the derivative measures how rapidly a function changes at a specific point.

If the limit of the difference quotient exists as $ \Delta x \to 0 $ and is finite, then the function is said to be differentiable at that point.

$$ \lim_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x} = l \in \mathbb{R} $$

If the limit does not exist or becomes infinite, the function is not differentiable at that point.

The process of computing this limit is called differentiation.

Example

The value of the limit of the difference quotient is called the first derivative and is denoted by f'(x).

Several equivalent notations are commonly used for the first derivative.

$$ f'(x)=Df=\frac{dy}{dx}=y' $$

The notation dy/dx is known as Leibniz notation.

From a geometric point of view, the derivative represents the slope of the tangent line to the graph of the function at the point $ x $.

Note. The first derivative f'(x) is itself a function and can be differentiated again. The derivative of the derivative is called the second derivative, denoted by f''(x). Repeating the process leads to higher-order derivatives.

Conditions for Differentiability

More formally, a function f(x) defined on the interval (a,b) is differentiable at a point x if the limit of the difference quotient exists as h approaches zero and the limit is finite.

A function is differentiable over an open interval (a, b) if it is differentiable at every point x strictly between a and b - that is, for all x ∈ (a, b).

A function is differentiable over a closed interval [a, b] if it is differentiable at every point x ∈ [a, b], and if the right-hand derivative exists at x = a and the left-hand derivative exists at x = b.

Points at which a function fails to be differentiable are called singular points.

Example: Computing a Derivative

Let’s work through the derivative of the function at a generic point \( x \).

\[ f(x)=x^3-x \]

By definition, the derivative is calculated using the difference quotient:

\[ f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \]

To compute \( f(x+h) \), replace every occurrence of \( x \) in the original function with \( x+h \).

\[ f(x + h ) = (x+h)^3-(x+h) \]

Now substitute the expressions for \( f(x) \) and \( f(x+h) \) into the formula:

\[ f'(x)=\lim_{h \to 0} \frac{[(x+h)^3-(x+h)]-(x^3-x)}{h} \]

The next step is to expand the cubic expression:

\[ (x+h)^3=x^3+3x^2h+3xh^2+h^3 \]

Substituting the expanded form gives:

\[ f'(x)=\lim_{h \to 0} \frac{x^3+3x^2h+3xh^2+h^3-x-h-x^3+x}{h} \]

After simplifying the numerator:

\[ f'(x)=\lim_{h \to 0} \frac{3x^2h+3xh^2+h^3-h}{h} \]

Every term in the numerator contains \( h \), so it can be factored out:

\[ f'(x)=\lim_{h \to 0} \frac{h(3x^2+3xh+h^2-1)}{h} \]

The factor \( h \) can now be simplified:

\[ f'(x)=\lim_{h \to 0} (3x^2+3xh+h^2-1) \]

As \( h \to 0 \), the terms containing \( h \) vanish:

\[f'(x)=3x^2-1 \]

This is the derivative of the function. By substituting different values of \( x \), you can calculate the derivative at any point of the domain.

The graph helps visualize the meaning of the derivative. The derivative describes how the slope of the tangent line changes from point to point along the curve.

Where the function \( f(x) \) is increasing, the derivative \( f'(x) \) is positive. Where the function is decreasing, the derivative is negative.

Note. For the function \( f(x)=x^3-x \), the tangent line has a positive slope on the intervals where the function increases, so the derivative satisfies \( f'(x)>0 \). On the intervals where the function decreases, the tangent line has a negative slope, and therefore \( f'(x)<0 \).

Left and Right Derivatives

Sometimes, a function may not be differentiable at a particular point x, but it may still be differentiable on either side of x.

In such cases, the finite limits taken from the left and the right are referred to as the left-hand derivative and the right-hand derivative at x, respectively.

What’s the difference?

For the left-hand derivative, the increment Δx approaches 0 from the negative side, meaning Δx < 0.

For the right-hand derivative, Δx approaches 0 from the positive side, so Δx > 0.

Right-hand and left-hand derivatives play a fundamental role in understanding whether a function is differentiable at a specific point.

If a function is differentiable at the point x, then both the right-hand and left-hand derivatives exist and have the same value. $$ f'(x) = f'_-(x)=f'_+(x) $$

If the function is not differentiable at that point, the one-sided derivatives may still exist, but they can produce different values. This commonly happens in functions with corners or cusps.

One important detail is often overlooked: one-sided derivatives exist only if the function itself is defined at the point.

If the function is not defined at \( x \), then neither the ordinary derivative nor the one-sided derivatives can exist.

A practical example

A classic example is the absolute value function:

\[ f(x)=|x| \]

We can rewrite the function in piecewise form by removing the absolute value notation:

\[ f(x)=|x| = \begin{cases} x & \text{if } x \geq 0 \\ \\  -x & \text{if } x < 0 \end{cases} \]

This function is defined for every real number and is also continuous at \( x=0 \).

However, the right-hand and left-hand derivatives at \( x=0 \) are different.

The right-hand derivative is:

\[ f'_+(0)=\lim_{h\to 0^+}\frac{|0+h|-|0|}{h}  =\lim_{h\to 0^+}\frac{|h|}{h} \]

Since \( h>0 \), we have \( |h|=h \). Therefore:

\[ f'_+(0)=\lim_{h\to 0^+}\frac{h}{h}=1 \]

The left-hand derivative is:

\[ f'_-(0)=\lim_{h\to 0^-}\frac{|0+h|-|0|}{h} =\lim_{h\to 0^-}\frac{|h|}{h} \]

Since \( h<0 \), we have \( |h|=-h \). Therefore:

\[ f'_-(0)=\lim_{h\to 0^-}\frac{-h}{h}=-1 \]

Because the right-hand and left-hand derivatives are different, \( f'_+(0)\neq f'_-(0) \), the function is not differentiable at \( x=0 \).

From a geometric point of view, the graph has a sharp corner at the origin.

 

Example 2

Now consider the function:

\[ f(x)=\frac{1}{x^2} \]

This function is not differentiable at \( x=0 \) because it is not defined there. Its domain is:

\[ D=\mathbb{R}\setminus{0} \]

Since the value \( f(0) \) does not exist, the derivative at \( x=0 \) cannot be defined either.

The derivative does exist at every other point in the domain:

\[ f'(x)=-\frac{2}{x^3} \]

There is, however, an important difference compared with the function \( \frac{1}{x} \).

For the function \( \frac{1}{x^2} \), the limit approaches \( +\infty \) from both sides:

\[ \lim_{x\to 0^-}\frac{1}{x^2}=+\infty \]

\[ \lim_{x\to 0^+}\frac{1}{x^2}=+\infty \]

This means the graph rises toward \( +\infty \) on both sides of the vertical asymptote \( x=0 \).

Do one-sided derivatives exist at x=0? No. Strictly speaking, neither the right-hand derivative nor the left-hand derivative exists at \( x=0 \). One-sided derivatives require the function to be defined at the point. $$ f'_+(x_0)=\lim_{h\to 0^+}\frac{f(x_0+h)-f(x_0)}{h} $$ $$ f'_-(x_0)=\lim_{h\to 0^-}\frac{f(x_0+h)-f(x_0)}{h} $$ For the function \( f(x)=\frac{1}{x^2} \), when \( x_0=0 \) the expression contains the term \( f(0) \), which is undefined. Therefore, \( \frac{f(h)-f(0)}{h} \) is also undefined. As a result, neither the right-hand derivative nor the left-hand derivative exists in the rigorous mathematical sense. It is important not to confuse one-sided derivatives with one-sided limits of a function. In the case of \( f(x)=\frac{1}{x^2} \), the one-sided limits are: \[ \lim_{x\to 0^-}f(x)=+\infty \] \[ \lim_{x\to 0^+}f(x)=+\infty \] These are limits of the function as \( x \) approaches \( 0 \), not one-sided derivatives at the point itself. They are two different mathematical concepts.

Derivative of a Function at a Point or on an Interval

The derivative of a function can be studied either at a single point or across an interval. The two ideas are closely connected, but they are used for different purposes in calculus.

• The derivative at a point

  Given a function \( f(x) \), the derivative at the point \( x_0 \) is defined as the limit of the difference quotient, provided that the limit exists and is finite: \[ f'(x_0)=\lim_{h\to 0}\frac{f(x_0+h)-f(x_0)}{h} \] When this limit exists, the function is said to be differentiable at the point \( x_0 \).

  The derivative at a point measures the instantaneous rate of change of the function at that specific location. In geometric terms, it corresponds to the slope of the tangent line to the graph of the function at the point considered.

  Example. The function \( f(x)=|x| \) is differentiable at the point \( x_0 = 2 \), but it is not differentiable at \( x_0 = 0 \). Although the function is defined at \( 0 \), the left-hand derivative and the right-hand derivative are different there. The function \( f(x)=\frac{1}{x^2} \) is differentiable at the point \( x_0 = 2 \), but it is not differentiable at \( x_0 = 0 \) because the function itself is not defined at that point.

• The derivative on an interval

  A function \( f(x) \) is said to be differentiable on an interval \( (a,b) \) if it has a derivative at every point \( x \) in the interval: \[ f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h} \] In this situation, the derivative defines a new function, called the derivative function, denoted by \( f'(x) \).

  For a closed interval \( [a,b] \), it is also necessary to verify that the right-hand derivative at \( a \) and the left-hand derivative at \( b \) both exist and are finite: \[ f'(a)=\lim_{h\to 0^+}\frac{f(a+h)-f(a)}{h} \] \[ f'(b)=\lim_{h\to 0^-}\frac{f(b+h)-f(b)}{h} \]

  When discussing differentiability on an interval, the focus is no longer on a single point but on the behavior of the function throughout an entire range of values. Studying the derivative over an interval makes it possible to determine where the function is increasing, where it is decreasing, and where it reaches relative maxima or minima.

  Example. The function \( f(x)=|x| \) is differentiable on the interval \( [0,5] \) because it has a derivative at every point of the open interval \( (0,5) \), together with a right-hand derivative at \( 0 \) and a left-hand derivative at \( 5 \).

  Note. However, some textbooks adopt a more restrictive definition of differentiability on a closed interval. According to this convention, the function must also be differentiable at the endpoints \( a \) and \( b \) in the ordinary sense, meaning that the derivative must be defined through the full two-sided limit even at the endpoints. Under this definition, the function \( f(x)=|x| \) would not be differentiable on the interval \( [0,5] \), because it is not differentiable at the endpoint \( 0 \).

In summary, the derivative at a point describes the local behavior of a function at a specific location, while the derivative on an interval describes how the function behaves across an entire continuous set of points through its derivative function.

Derivative Corollaries

• If a function is differentiable at x and its derivative f'(x) is finite, then the left-hand and right-hand derivatives not only exist, but are equal to the derivative:

  $$ \text{If} \:\: f'(x) = l \:\Rightarrow \: f'(x) = f'_-(x) = f'_+(x) $$

  On the other hand, if the function is not differentiable at x, then the left-hand and right-hand derivatives may differ or one of them might not exist at all.

Leibniz Notation

One of the most common notations used in calculus is Leibniz's notation.

$$ \frac{dy}{dx} $$ or $$ \frac{δy}{δx} $$

The numerator represents the dependent variable (the function’s output).

The denominator specifies the independent variable with respect to which the differentiation is performed.

Note. Leibniz’s notation is especially helpful when dealing with functions of multiple variables and it’s important to indicate which variable you're differentiating with respect to. If only one variable is involved, it’s often more convenient to use a more compact notation such as y''.

And so on.