Derivatives and Derivative Rules
The derivative is the central tool of differential calculus. It measures how steeply the graph of a function rises or falls — the slope of the function — at each point of its domain. This page follows Walz, Foundations of Mathematics, Section 6.1: it builds the derivative from the intuitive notion of slope, gives its precise definition as a limit of difference quotients, and then develops the rules that let us differentiate large classes of functions without returning to that limit each time.
The Slope of a Function
Imagine a hike through mountainous terrain, modelled as a function $f : \lbrack a, b \rbrack \to \mathbb{R}$ that assigns to each location $x$ a height $f(x)$. The steepness we must overcome depends on where we are: the slope is generally not constant but varies with $x$. These varying slope values are themselves described by a new function, the first derivative of $f$, written $f'$ (read "f dash").
Before giving the general definition, we compute the derivatives of a few simple functions directly.
Constant Functions
A constant function $f(x) = c$ has a horizontal graph, so its slope is $0$ everywhere.
First derivative of a constant function. For $f(x) = c$ the first derivative vanishes for all $x$: $f'(x) = 0.$
Linear Functions
For a linear function $f(x) = cx + d$ the graph is a straight line, whose slope is constant. Moving a small step $h \neq 0$ from $x$ to $x + h$ changes the value by $f(x+h) - f(x) = c\,h$, so the ratio of the change in value to the change in position is $\frac{f(x+h) - f(x)}{(x+h) - x} = \frac{c\,h}{h} = c.$
First derivative of a linear function. For $f(x) = cx + d$ the first derivative is the constant $c$: $f'(x) = c.$
In particular the identity $f(x) = x$ (with $c = 1$, $d = 0$) has $f'(x) = 1$.
The Quadratic Power Function
For $f(x) = x^2$ the slope is not constant. Using the binomial formula, the ratio of the change in value to the change in position between $x$ and $x + h$ is $\frac{f(x+h) - f(x)}{(x+h) - x} = \frac{(x+h)^2 - x^2}{h} = \frac{2hx + h^2}{h} = 2x + h.$
This ratio still depends on $h$: it is the slope of the secant line through the two points $(x \mid x^2)$ and $(x+h \mid (x+h)^2)$. To obtain the actual slope at $x$ we let $h$ shrink towards $0$ (written $h \to 0$), which sends the secant into the tangent line. Taking this limit gives $\lim_{h \to 0} (2x + h) = 2x.$
First derivative of the quadratic power function. For $f(x) = x^2$: $f'(x) = 2x.$
The General Power Function
The same pattern holds for every power $p_i(x) = x^i$ with $i \in \mathbb{N}$: bring the exponent down as a factor and reduce it by one.
First derivative of the power function. For $p_i(x) = x^i$: $p_i'(x) = i\,x^{i-1}.$
For example $(x^{128})' = 128\,x^{127}$ and $(x^7)' = 7 x^6$. As we will see below, this "rough rule" also extends to negative and fractional exponents.
The Derivative as a Limit
We now make the procedure above precise for a general function $f$. Fix a point $x$ and a small $h \neq 0$ such that both $x$ and $x + h$ lie in the domain of $f$.
- The secant line of $f$ through $x$ and $x + h$ is the straight line through $(x \mid f(x))$ and $(x+h \mid f(x+h))$.
- Its slope is the difference quotient $\frac{f(x+h) - f(x)}{h}.$
The difference quotient is only an approximation to the actual slope at $x$. Passing to the limit $h \to 0$ gives the exact value.
Differential quotient and first derivative. If the limit of the difference quotients $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ exists, then $f'(x)$ is the value of the slope of $f$ at $x$, called the first derivative of $f$ at $x$. In that case $f$ is said to be differentiable at $x$.
Tangent of $f$ at $x$. The linear function through $(x \mid f(x))$ with slope $f'(x)$ is the tangent line of $f$ at $x$: $t(\bar{x}) = f'(x)\,\bar{x} - f'(x)\,x + f(x).$
If $f$ is differentiable at every point of its domain, the assignment $x \mapsto f'(x)$ defines a new function, the derivative function (or simply the derivative) of $f$.
Derivatives of Standard Functions
The limit procedure yields the following derivatives of the standard functions. Each can be derived from the definition using the appropriate limit laws.
- The exponential function is its own derivative: for $\exp(x) = e^x$, $\exp'(x) = e^x.$ The key step uses $\lim_{h \to 0} \dfrac{e^h - 1}{h} = 1$.
- The sine and cosine functions satisfy $\sin'(x) = \cos(x), \qquad \cos'(x) = -\sin(x).$
- The root function $f(x) = \sqrt{x}$ on an interval $\lbrack a, b \rbrack$ with $a > 0$ has $f'(x) = \frac{1}{2\sqrt{x}}.$ Equivalently $(x^{1/2})' = \tfrac{1}{2} x^{-1/2}$, so the power rule extends to fractional exponents.
When Differentiation Fails
Not every function is differentiable. The limit of the difference quotients can fail to exist in two characteristic ways.
- The root function $f(x) = \sqrt{x}$ is not differentiable at $x = 0$: as $x \to 0^+$ the slope grows beyond every bound, $\lim_{h \to 0^+} \tfrac{1}{\sqrt{h}} = \infty$. It is differentiable at every $x > 0$.
- The absolute value function $f(x) = \lvert x \rvert$ is not differentiable at $x = 0$: approaching from the right gives slope $+1$, from the left gives $-1$. With two conflicting candidates the limit cannot be uniquely determined, and the graph shows a "kink".
Differentiable functions, by contrast, have smooth graphs without kinks or vertical tangents.
Derivative Rules
Returning to difference quotients for every function is tedious. The following rules let us differentiate combinations of known functions directly.
Factor Rule
For a differentiable $f$ and a real constant $\lambda$, the product $\lambda \cdot f$ is differentiable and $(\lambda f)'(x) = \lambda\,f'(x).$
For example $(4\sqrt{x})' = 4 \cdot \tfrac{1}{2\sqrt{x}} = \tfrac{2}{\sqrt{x}}$.
Sum and Difference Rules
For differentiable $f$ and $g$, the sum $f + g$ is differentiable and $(f + g)'(x) = f'(x) + g'(x).$
Writing $f - g = f + (-1)\cdot g$ and combining with the factor rule gives the difference rule $(f - g)'(x) = f'(x) - g'(x).$
Derivative of polynomial functions. Applying the factor and sum rules to a polynomial $p(x) = a_n x^n + \cdots + a_1 x + a_0$ gives the polynomial of degree $n - 1$: $p'(x) = n\,a_n x^{n-1} + \cdots + i\,a_i x^{i-1} + \cdots + a_1.$
For example $p(x) = 3x^5 - \sqrt{2}\,x^3 + \tfrac{8}{7} x^2 - 3$ has $p'(x) = 15 x^4 - 3\sqrt{2}\,x^2 + \tfrac{16}{7} x$.
Product Rule
The derivative of a product is not the product of the derivatives. Instead:
Product rule. For differentiable $f$ and $g$, the product $f \cdot g$ is differentiable and $(f \cdot g)'(x) = f'(x)\,g(x) + g'(x)\,f(x).$
For example, with $f(x) = g(x) = \sin(x)$ we get $\bigl(\sin^2(x)\bigr)' = 2\cos(x)\sin(x)$.
Chain Rule
To differentiate a composition $f \circ g$ (first apply $g$, then $f$):
Chain rule. For differentiable $f$ and $g$, the composition $f \circ g$ is differentiable and $(f \circ g)'(x) = g'(x)\,f'\bigl(g(x)\bigr).$
For example, with $g(x) = 3x^2 + 1$ and $f(\bar{x}) = \bar{x}^{71}$: $\bigl((3x^2+1)^{71}\bigr)' = 6x \cdot 71\,(3x^2+1)^{70} = 426x\,(3x^2+1)^{70}.$ Likewise $\bigl(e^{\sin(x)}\bigr)' = \cos(x)\,e^{\sin(x)}$.
Applying the product rule to $x \cdot x^{-1} = 1$ and solving gives the derivative of $x^{-1}$: $\bigl(x^{-1}\bigr)' = -x^{-2} = -\frac{1}{x^2},$ confirming that the power rule holds for negative exponents as well.
Quotient Rule
Combining the product and chain rules yields the rule for quotients, valid wherever the denominator does not vanish.
Quotient rule. For differentiable $f$ and $g$ with $g(x) \neq 0$: $\left(\frac{f}{g}\right)'(x) = \frac{f'(x)\,g(x) - g'(x)\,f(x)}{\bigl(g(x)\bigr)^2}.$
A classic application is the tangent function $\tan(x) = \dfrac{\sin(x)}{\cos(x)}$ for $x \in \left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$: $\tan'(x) = \frac{\cos^2(x) + \sin^2(x)}{\cos^2(x)} = 1 + \tan^2(x).$
Where This Leads
With these rules we can differentiate essentially every function built from powers, roots, exponentials, and trigonometric functions. The next page, Applications of Derivatives and Extrema, uses the first derivative to locate the largest and smallest values of a function and to determine where it rises and falls.