Derivatives

In this chapter we treat derivatives as the central tool of differential calculus: a systematic way to measure and use instantaneous rates of change.

The parent chapter (“Limits and Continuity”) has already introduced limits and the intuitive idea of a derivative (slope of a tangent, instantaneous rate of change). Here we turn that intuition into precise, workable formulas and methods. The next chapter (“Applications of Derivatives”) will use what you learn here; we will only hint at applications when it helps understanding.

1. The derivative as a function

Previously, you may have seen the derivative of a function $f$ at a single point $x=a$, written $f'(a)$, defined using a limit. In this chapter we focus on the derivative as a new function built from $f$.

Given a function $f(x)$, its derivative $f'(x)$ is defined (where it exists) by
$$
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
$$

Think of this as:

• For each $x$, we look at the slopes of secant lines over shorter and shorter intervals of length $h$, and
• We take the limit as $h$ approaches $0$.

Where this limit exists, the function is called differentiable at $x$, and the value $f'(x)$ is the instantaneous rate of change of $f$ at $x$.

The result is a function $f'$ that assigns to each input $x$ the slope of the tangent line at that point (if it exists).

2. Notation for derivatives

Several notations are used for the derivative; you should be comfortable with all of them, since different texts and contexts prefer different styles.

2.1 Prime notation

If $y = f(x)$, the derivative of $f$ with respect to $x$ is
$$
f'(x)
$$
read as “$f$ prime of $x$”. Higher derivatives use more primes:

• Second derivative: $f''(x)$
• Third derivative: $f'''(x)$
• Beyond that: $f^{(4)}(x), f^{(5)}(x), \dots$

2.2 Leibniz notation

If $y$ is a function of $x$, we often write
$$
\frac{dy}{dx}
$$
for the derivative of $y$ with respect to $x$. If we want to emphasize the function $f$,
$$
\frac{df}{dx}, \quad \frac{d}{dx}f(x)
$$
all mean the same quantity, when $f$ depends on $x$.

Leibniz notation is especially useful when:

• Indicating the variable we are differentiating with respect to, e.g. $\dfrac{d}{dt}$ vs. $\dfrac{d}{dx}$.
• Using rules like the chain rule, product rule, etc., which look natural in this notation.
• Working with related rates (in the next chapter), because it resembles fractions and helps track units.

Higher derivatives:
$$
\frac{d^2 y}{dx^2}, \quad \frac{d^3 y}{dx^3}, \quad \frac{d^n y}{dx^n}.
$$

2.3 Other notations

You might also see:

• Dot notation (often for time derivatives in physics):
  $$
  \dot{y} = \frac{dy}{dt}, \quad \ddot{y} = \frac{d^2 y}{dt^2}.
  $$
• Subscript notation when there are many variables:
  $$
  f_x \quad \text{meaning “partial derivative of $f$ with respect to $x$”}
  $$
  (this belongs to multivariable calculus; here we only work with single-variable derivatives).

In this chapter, the most common will be $f'(x)$ and $\frac{dy}{dx}$.

3. Computing derivatives from the definition

Before learning general rules, it is important to see how the limit definition works in concrete examples.

This process is:

1. Start with
   $$
   f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}.
   $$
2. Simplify the expression inside the limit.
3. Cancel $h$ (if possible) so that the limit can be evaluated as $h\to 0$.
4. Take the limit.

3.1 Example: $f(x) = x^2$

Let $f(x) = x^2$.
$$
\begin{aligned}
f'(x)
&= \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} \\
&= \lim_{h\to 0} \frac{(x+h)^2 - x^2}{h} \\
&= \lim_{h\to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \\
&= \lim_{h\to 0} \frac{2xh + h^2}{h} \\
&= \lim_{h\to 0} (2x + h) \\
&= 2x.
\end{aligned}
$$

So $f'(x) = 2x$. In other words, the slope of the tangent to $y = x^2$ at $x$ is $2x$.

3.2 Example: $f(x) = \frac{1}{x}$ (for $x\ne 0$)

Let $f(x) = x^{-1}$.
$$
\begin{aligned}
f'(x)
&= \lim_{h\to 0} \frac{f(x+h) - f(x)}{h} \\
&= \lim_{h\to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \\
&= \lim_{h\to 0} \frac{\frac{x - (x+h)}{x(x+h)}}{h} \\
&= \lim_{h\to 0} \frac{\frac{-h}{x(x+h)}}{h} \\
&= \lim_{h\to 0} \frac{-1}{x(x+h)} \\
&= -\frac{1}{x^2}.
\end{aligned}
$$

So $f'(x) = -\dfrac{1}{x^2}$ for $x\ne 0$.

3.3 When the limit does not exist

The limit may fail to exist, and in that case the derivative does not exist. The limit might fail because:

• The left-hand and right-hand limits are different.
• The expression becomes unbounded.
• The function is not defined in a way that the limit makes sense.

You saw examples of non-differentiable behavior in the discussion of continuity and corners; here we connect that to the formal definition in the next section.

4. Differentiability and where derivatives fail

A function $f$ is differentiable at $x=a$ if the limit
$$
\lim_{h\to 0} \frac{f(a+h) - f(a)}{h}
$$
exists (as a finite number). If $f$ is differentiable at every point in an interval, we say $f$ is differentiable on that interval.

Some important facts:

• If $f$ is differentiable at $a$, then it is continuous at $a$.
• The converse is not true: continuity does not guarantee differentiability.

4.1 Typical ways differentiability fails

1. Corners and cusps: The left-hand slope and the right-hand slope at a point are different or infinite.
• Example: $f(x) = |x|$ at $x=0$.
  For $x>0$, $f(x) = x$ so slope is $.
  For $x<0$, $f(x) = -x$ so slope is $-1$.
  The one-sided derivatives differ, so the derivative does not exist at
$.
2. Vertical tangent lines: The tangent line is vertical, which corresponds to infinite slope.
• Example: $f(x)=\sqrt[3]{x}$ has a vertical tangent at $x=0$ (its derivative there blows up).
3. Discontinuity: If the function is not continuous at $a$, it cannot be differentiable at $a$.

Recognizing these situations is important; they are common in graphs and real-world models.

5. Basic derivative rules (without proofs)

Derivatives from the definition are often messy. Fortunately, we can use general rules to compute derivatives efficiently. A later chapter on differentiation rules will go into full detail and practice; here we state the most basic ones that you will use constantly.

Let $f(x)$ and $g(x)$ be differentiable functions at $x$, and let $c$ be a constant.

5.1 Constant rule

If $f(x) = c$ (a constant), then
$$
f'(x) = 0.
$$

5.2 Power rule (for integer powers)

For any integer $n$,
$$
\frac{d}{dx}\left( x^n \right) = n x^{n-1}.
$$

This includes:

• $n=1$: $\dfrac{d}{dx}(x) = 1$,
• $n=2$: $\dfrac{d}{dx}(x^2) = 2x$,
• $n=3$: $\dfrac{d}{dx}(x^3) = 3x^2$,
• $n=-1$: $\dfrac{d}{dx}(x^{-1}) = -x^{-2}$.

Later the power rule will be extended to many non-integer exponents.

5.3 Constant multiple rule

If $f$ is differentiable and $c$ is a constant,
$$
\frac{d}{dx}\big( c f(x) \big) = c f'(x).
$$

5.4 Sum and difference rules

If $f$ and $g$ are differentiable,
$$
\frac{d}{dx}\big( f(x) + g(x)\big) = f'(x) + g'(x),
$$
$$
\frac{d}{dx}\big( f(x) - g(x)\big) = f'(x) - g'(x).
$$

These basic rules allow you to differentiate any function built from constant multiples, powers of $x$, and sums or differences.

More advanced rules (product rule, quotient rule, chain rule, etc.) belong to the “Differentiation rules” section, not here, though we will briefly preview them in words later.

6. Derivatives of common elementary functions

Using the limit definition (and techniques from earlier chapters), one can prove standard derivative formulas for basic functions. Here we simply list them, since they are used constantly and are considered fundamental.

In all formulas, $x$ is in a domain where the function is defined and differentiable.

6.1 Powers of $x$

For any integer $n$,
$$
\frac{d}{dx}\left( x^n \right) = n x^{n-1}.
$$

(As noted, later this extends to many real $n$.)

6.2 Exponential functions with base $e$

The natural exponential function has the special property:
$$
\frac{d}{dx}(e^x) = e^x.
$$

This means the rate of change of $e^x$ is equal to its own value at every point.

For a general exponential $a^x$ (with $a>0$, $a\ne 1$),
$$
\frac{d}{dx}(a^x) = a^x \ln a,
$$
where $\ln a$ is the natural logarithm of $a$.

6.3 Logarithmic functions

The natural logarithm:
$$
\frac{d}{dx}(\ln x) = \frac{1}{x}, \quad x>0.
$$

For logarithm base $a$,
$$
\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}, \quad x>0.
$$

6.4 Trigonometric functions (in radians)

Using $x$ measured in radians, the key derivatives are:

• Sine and cosine:
  $$
  \frac{d}{dx}(\sin x) = \cos x,
  $$
  $$
  \frac{d}{dx}(\cos x) = -\sin x.
  $$
• Tangent:
  $$
  \frac{d}{dx}(\tan x) = \sec^2 x.
  $$
• Cotangent, secant, cosecant:
  $$
  \frac{d}{dx}(\cot x) = -\csc^2 x,
  $$
  $$
  \frac{d}{dx}(\sec x) = \sec x \tan x,
  $$
  $$
  \frac{d}{dx}(\csc x) = -\csc x \cot x.
  $$

Derivatives of inverse trigonometric and hyperbolic functions are usually listed later in a more advanced differentiation-rules section.

6.5 Examples combining basic rules and common derivatives

1. $f(x) = 3x^2 - 5x + 7$.
   $$
   f'(x) = 3\cdot 2x - 5\cdot 1 + 0 = 6x - 5.
   $$
2. $g(x) = 4e^x - \frac{1}{x}$.
   $$
   g'(x) = 4e^x - \left(-\frac{1}{x^2}\right) = 4e^x + \frac{1}{x^2}, \quad x\ne 0.
   $$
3. $h(x) = \sin x + 2\cos x$.
   $$
   h'(x) = \cos x - 2\sin x.
   $$

Here we used only the constant multiple rule, sum rule, and simple derivatives from the list.

7. The derivative as slope and as rate of change

Although this chapter focuses on formulas, it is important to keep two key interpretations in mind.

7.1 Slope of the tangent line

If $y = f(x)$ and $f$ is differentiable at $x=a$, then
$$
f'(a)
$$
is the slope of the tangent line to the graph of $y = f(x)$ at the point $(a, f(a))$.

The equation of the tangent line is
$$
y - f(a) = f'(a)(x - a).
$$

This connects the derivative to straight-line approximations of curves; later, when studying applications and more advanced topics, this idea of “local linearity” becomes central.

7.2 Instantaneous rate of change

If a quantity $y$ depends on another quantity $x$, the derivative
$$
\frac{dy}{dx}
$$
is the instantaneous rate of change of $y$ with respect to $x$. Roughly:

• If $x$ changes a tiny amount $\Delta x$,
• Then $y$ changes approximately by $f'(x)\Delta x$.

In other words, for small changes,
$$
\Delta y \approx f'(x)\Delta x.
$$

This linear approximation underlies many practical uses of derivatives in physics (velocity, acceleration), economics (marginal cost, marginal revenue), biology, engineering, and so on. Detailed word problems and applications will appear in the “Applications of Derivatives” chapter.

8. Higher-order derivatives

If $f$ is differentiable and we form its derivative $f'$, we can sometimes differentiate again.

• The second derivative is
  $$
  f''(x) = \frac{d}{dx}\big(f'(x)\big) = \frac{d^2 y}{dx^2}.
  $$
• The third derivative:
  $$
  f'''(x) = \frac{d}{dx}\big(f''(x)\big) = \frac{d^3 y}{dx^3}.
  $$
• In general, the $n$-th derivative:
  $$
  f^{(n)}(x) = \frac{d^n y}{dx^n}.
  $$

Higher derivatives, when they exist, give more detailed information about how the function changes:

• First derivative: rate of change (slope, velocity).
• Second derivative: rate of change of the first derivative (acceleration, concavity).
• Higher derivatives: further changes-of-change, often appearing in physical models and differential equations.

Graphical interpretations and uses of second derivatives (concavity, inflection points, etc.) will be developed in the “Curve sketching” and “Applications of Derivatives” sections.

9. A brief preview of more advanced rules

This chapter has treated the derivative mainly through its definition and the most basic rules. In practice, you will also need:

• Product rule: derivative of a product $f(x)g(x)$.
• Quotient rule: derivative of $\dfrac{f(x)}{g(x)}$.
• Chain rule: derivative of a composition $f(g(x))$.

These rules, with their proofs and many examples, are the focus of the “Differentiation rules” section that follows. Here you only need to know that such rules exist and that they allow differentiation of far more complicated combinations of functions.

10. Summary of key points

• The derivative of $f$ at $x$ is defined by
  $$
  f'(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h},
  $$
  when this limit exists.
• The derivative is itself a function, giving a slope or rate of change at each point.
• There are several common notations: $f'(x)$, $\dfrac{dy}{dx}$, $\dfrac{d}{dx}f(x)$.
• Differentiability implies continuity, but not vice versa.
• Basic rules: constant rule, power rule (for integer powers), constant multiple rule, sum/difference rules.
• Fundamental derivatives to know:
• $\dfrac{d}{dx}(x^n) = n x^{n-1}$,
• $\dfrac{d}{dx}(e^x) = e^x$, $\dfrac{d}{dx}(a^x) = a^x\ln a$,
• $\dfrac{d}{dx}(\ln x) = \dfrac{1}{x}$,
• $\dfrac{d}{dx}(\sin x) = \cos x$, $\dfrac{d}{dx}(\cos x) = -\sin x$, etc.
• The derivative has two main interpretations:
• Slope of the tangent line to a graph.
• Instantaneous rate of change of one quantity with respect to another.
• Higher derivatives ($f''(x)$, $f^{(n)}(x)$) describe rates of change of rates of change, and are widely used in modeling and analysis.

These ideas form the computational and conceptual backbone that the next chapters will use to explore both powerful rules for differentiation and many applications.