Introduction to Derivatives: Meaning, Rules, Examples

What Is A Derivative?

The first fact in any introduction to derivatives is this: a derivative is the rate at which a function's output changes as its input changes, measured at a single input value. Geometrically, it is the slope of the line that just touches the graph at that point. For a function $f$, the derivative at the input $a$ is written $f'(a)$ and read "f prime of a".

The formal definition uses a limit, the value an expression approaches as one quantity gets closer and closer to a target:

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

• $f$ is the function, and $a$ is the input you care about.

• $h$ is a small step away from $a$. It can be positive or negative, but it is never exactly $0$ inside the fraction.

• $f(a+h) - f(a)$ is the change in output over that step.

• The fraction is the slope of the line through two points on the graph. The limit asks where that slope settles as the step shrinks.

The derivative exists only when this limit exists and is a finite number. This page gives you the picture and the first tools. The full limit machinery has its own pages: the definition of the derivative unpacks the formula line by line, and the limit definition of derivative works through many first-principles problems. Once $f'(a)$ is found at every input, the result is a new function, the derivative of a function, usually written $f'(x)$.

How Does A Derivative Measure Slope?

A derivative measures slope by turning a curve into a straight line at one point. For a straight line, slope is rise over run, and it is the same everywhere. A curve has no single slope, so the derivative zooms in until the curve near one point looks straight.

Draw a line through two points on the curve, $(a, f(a))$ and $(a+h, f(a+h))$. That line is a secant line, and its slope is the fraction inside the limit.

Slide the second point toward the first and the secant line swings around until it settles on the tangent line, which touches the curve at $(a, f(a))$ and runs in the curve's direction there. The derivative $f'(a)$ is the slope of that tangent line. Writing the line's full equation is its own skill, covered in tangent line equations.

How Does Average Rate Of Change Become Instantaneous?

The average rate of change becomes instantaneous when you shrink the time interval toward zero and watch where the average settles. That settled value is the derivative. This answers a question people ask on Quora: if the purpose of the derivative is to find a rate of change, why not just use the rate of change you already know?

The rate you already know is the average rate of change: total change divided by the length of the interval. It answers "how fast on average?", not "how fast right now?". A stone dropped from rest falls about $d(t) = 4.9t^2$ metres after $t$ seconds (air resistance ignored). Here is its average speed over shorter and shorter intervals starting at $t = 3$:

The averages close in on $29.4$. Algebra confirms it, with each step on its own line:

$$\frac{d(3+h) - d(3)}{h} = \frac{4.9(9 + 6h + h^2) - 44.1}{h}$$

$$= \frac{29.4h + 4.9h^2}{h}$$

$$= 29.4 + 4.9h$$

As $h \to 0$, the term $4.9h$ vanishes, so $d'(3) = 29.4,\text{m/s}$. That number is the stone's speed at the instant $t = 3$. No interval of zero length was ever used. The limit describes where the averages are heading, which is how a speedometer can honestly show a speed "right now".

Is A Derivative The Same As Slope?

A derivative is the same as slope only for straight lines. For a curve, the derivative gives the slope of the tangent line at each point, so it is a slope that changes from point to point. Versions of this question come up again and again on Quora ("Are derivatives and slopes the same thing?") and Physics Forums ("Is the derivative the same as the slope of a function?").

• For a line: $f(x) = 3x + 5$ has slope $3$ everywhere, and its derivative is $f'(x) = 3$ at every input. The two ideas agree.

• For a curve: $f(x) = x^2$ has no single slope. Its derivative $f'(x) = 2x$ is a formula that hands back a different tangent slope at each $x$: $-2$ at $x = -1$, $0$ at $x = 0$, $6$ at $x = 3$.

So "slope" is a number that belongs to a line, while the derivative is a function that belongs to a curve. The derivative gives each point of the curve its own slope.

What Notation Do Derivatives Use?

Derivatives have several notations because calculus was invented twice, by people who wrote it differently. The three you will meet most are shown below. This article uses the prime notation $f'(x)$ throughout.

All four ask the same question: how fast is the output changing per unit of input? The $\frac{dy}{dx}$ form is not a real fraction. It stands for the limit of $\frac{\Delta y}{\Delta x}$, though it behaves like a fraction often enough that the chain rule later feels natural. The notation for differentiation has a longer history than most students expect.

How Do You Find A Derivative?

You find a derivative in one of two ways: from the limit definition, or with shortcut rules that were themselves proved from that definition. A beginner should do the first a few times and then switch to the second. Actually, a better guide than "a few times" is to keep using the definition until cancelling the $h$ feels routine, and only then trust the shortcuts.

The definition route has four steps:

1. Write $f(a+h)$ by replacing every $x$ with $a + h$.

2. Subtract $f(a)$ and simplify the numerator.

3. Divide by $h$, cancelling the $h$ before going further.

4. Let $h \to 0$ in what is left.

Step 3 is where most of the work hides. Until the $h$ in the denominator cancels, you cannot let $h$ reach $0$, because the fraction would become $\frac{0}{0}$. Example 2 below shows what happens when that step is skipped.

What Are The First Derivative Rules?

The first derivative rules turn the limit calculation into a one-line answer for powers, constants and sums. Four rules handle every polynomial:

Each rule has a short reason behind it:

• Constant rule: a constant's graph is a horizontal line, and a horizontal line has slope $0$. Nothing changes, so the rate of change is zero.

• Power rule: for $x^2$, the limit gives $\frac{(x+h)^2 - x^2}{h} = 2x + h$, which tends to $2x$. The same expansion for $x^3$ leaves $3x^2$. The power rule page proves the pattern for every exponent $n$, including negatives and fractions.

• Constant multiple rule: stretching a graph vertically by $c$ multiplies every slope by $c$. The constant multiple rule page gives the one-line proof.

• Sum rule: the change in $f + g$ is the change in $f$ plus the change in $g$, and limits respect addition.

Beyond polynomials, a handful of standard results come up early. They are previewed here, not derived, because each has its own lesson:

The trig pair is worked out in differentiation of trigonometric functions. Notice the radian condition: the neat result $\cos x$ holds only when angles are measured in radians.

What Are First And Second Order Derivatives?

A first order derivative $f'(x)$ measures how fast $f$ changes. A second order derivative $f''(x)$ measures how fast $f'$ itself changes, so it is the rate of change of the rate of change. For $f(x) = x^3$, the first derivative is $3x^2$ and the second is $6x$.

Motion gives both orders a physical meaning. If $d(t)$ is position, $d'(t)$ is velocity and $d''(t)$ is acceleration.

For the falling stone, $d'(t) = 9.8t$ and $d''(t) = 9.8$, the familiar acceleration due to gravity in $\text{m/s}^2$. The link between these three quantities is developed in velocity and acceleration. Third and higher derivatives exist too (physicists call the third derivative of position "jerk"), and they will matter again when you meet Taylor series.

When Does A Derivative Not Exist?

A derivative does not exist at a point where the secant slopes fail to settle on one finite number. Three shapes cause this, and a picture makes each one obvious:

• A corner: $f(x) = |x|$ at $x = 0$. Secant slopes from the left are $-1$ and from the right are $+1$. Two different answers means no single tangent slope.

• A vertical tangent: $f(x) = \sqrt[3]{x}$ at $x = 0$. The secant slopes grow without bound, and a vertical line has no finite slope.

• A break: at a jump or a hole in the graph, there is no single point for the tangent line to touch.

A function that has a derivative at a point is called differentiable there. Every differentiable function is continuous, but a continuous function can still have a corner. The full conditions are worked through in differentiability of a function.

Examples Of Derivatives

Example 1

Find the derivative of $f(x) = x^3$ and its value at $x = 2$.

Apply the power rule with $n = 3$:

$$f'(x) = 3x^{3-1}$$

$$f'(x) = 3x^2$$

Substitute $x = 2$:

$$f'(2) = 3(2)^2 = 12$$

Final answer: $f'(x) = 3x^2$, and the tangent slope at $x = 2$ is $12$.

Example 2

Use the limit definition to find $f'(3)$ for $f(x) = x^2$.

The tempting first move is to put $h = 0$ straight into the difference quotient:

$$\frac{(3+0)^2 - 9}{0} = \frac{0}{0}$$

That expression has no value. It is not $0$, and it is not "undefined, so no derivative". The definition says $h$ approaches $0$, never that $h$ equals $0$, so substituting first throws away the information the limit needs.

The correct route simplifies before taking the limit:

$$\frac{(3+h)^2 - 9}{h} = \frac{9 + 6h + h^2 - 9}{h}$$

$$= \frac{6h + h^2}{h}$$

$$= 6 + h$$

Now let $h \to 0$:

$$f'(3) = \lim_{h \to 0} (6 + h) = 6$$

The power rule agrees: $f'(x) = 2x$, so $f'(3) = 6$.

Final answer: $f'(3) = 6$.

Example 3

Differentiate $f(x) = 5x^2 - 2x + 6$ and find the slope at $x = -1$.

Differentiate term by term with the sum, constant multiple, power and constant rules:

$$f'(x) = 5(2x) - 2(1) + 0$$

$$f'(x) = 10x - 2$$

Substitute $x = -1$:

$$f'(-1) = 10(-1) - 2 = -12$$

Final answer: $f'(x) = 10x - 2$, and the slope at $x = -1$ is $-12$, so the graph is falling steeply there.

Example 4

Find $f'(x)$ for $f(x) = \frac{4}{x}$, then evaluate $f'(2)$.

Rewrite the fraction as a power before using the power rule:

$$f(x) = 4x^{-1}$$

$$f'(x) = 4(-1)x^{-2}$$

$$f'(x) = -\frac{4}{x^2}$$

Substitute $x = 2$:

$$f'(2) = -\frac{4}{4} = -1$$

Final answer: $f'(x) = -\frac{4}{x^2}$ for $x \neq 0$, and $f'(2) = -1$.

The first instinct on $\frac{4}{x}$ is to differentiate the top and bottom separately, which gives $\frac{0}{1} = 0$ and claims the curve is flat everywhere. Rewriting as $4x^{-1}$ first is the habit that removes that error.

Example 5

A stone falls $d(t) = 4.9t^2$ metres in $t$ seconds. Find its speed at $t = 4$ s and compare it with its average speed over the first 4 seconds.

Differentiate to get the velocity function:

$$d'(t) = 4.9(2t) = 9.8t$$

Speed at the instant $t = 4$:

$$d'(4) = 9.8(4) = 39.2,\text{m/s}$$

Average speed over $0$ to $4$ s:

$$\frac{d(4) - d(0)}{4 - 0} = \frac{78.4 - 0}{4} = 19.6,\text{m/s}$$

Final answer: the stone is moving at $39.2,\text{m/s}$ at $t = 4$ s, twice its average speed of $19.6,\text{m/s}$ over the first 4 seconds. The gap exists because the stone keeps speeding up.

Example 6

Find the point where the tangent line to $f(x) = x^2 - 4x + 1$ is horizontal.

A horizontal tangent has slope $0$, so set the derivative equal to $0$:

$$f'(x) = 2x - 4$$

$$2x - 4 = 0$$

$$x = 2$$

Find the height of the curve there:

$$f(2) = 4 - 8 + 1 = -3$$

Final answer: the tangent is horizontal at $(2, -3)$, the lowest point of this parabola. This reverse use of the derivative, solving $f'(x) = 0$, is the start of optimisation.

Why Did Mathematicians Need Derivatives?

Derivatives were invented to solve two old problems that turned out to be the same problem: "find the tangent, find the speed." Before calculus, mathematicians could find tangents to circles and a few special curves, and they could compute average speeds. Neither method worked for a general curve or a changing speed.

• The tangent problem. Gottfried Wilhelm Leibniz (1646 to 1716, Germany) published his differential calculus in 1684 in the journal Acta Eruditorum. The paper's title promised a new method for maxima, minima and tangents, and it contained the $d$ notation and rules for powers, products and quotients.

• The motion problem. Isaac Newton (1643 to 1727, England) laid the foundations of his method of fluxions in 1665 to 1666, while his university was closed by plague. He treated quantities as flowing in time, so his derivative was a velocity. His main text on the method was written in 1671 but not printed until 1736.

The two problems collapse into one because the speed of a moving object is the slope of the tangent to its distance-time graph. That is the shape of every derivative in this article: a rate of change and a slope, read off the same limit.

Where Are Derivatives Used In Real Life?

Derivatives are used wherever someone needs a rate at a moment rather than an average. On Quora, "What is a derivative and how is it used?" is one of the most common ways the question gets asked. Here are the main answers:

• Motion and safety: speed is the derivative of position, and acceleration is the derivative of speed. Braking distances, airbag triggers and rocket guidance all depend on rates at an instant.

• Business: marginal cost, the extra cost of producing one more unit, is the derivative of the cost function. It tells a factory whether the next batch is worth making.

• Best values: the lowest point of a parabola in Example 6 came from setting $f'(x) = 0$. The same move finds the cheapest can design, the fastest route and the strongest beam for a given amount of material.

• Growth and medicine: the derivative of a population or a drug concentration gives the growth or clearance rate at a given hour.

• Machine learning: training a model means nudging its settings downhill on an error surface, and "downhill" is read from derivatives (the method is called gradient descent).

Each of these is the "how fast right now?" question from the speedometer. The rate-of-change reading of the derivative is developed further in derivative as rate of change.

Where Do Students Trip Up On Derivatives?

Most early derivative errors come from treating the derivative as a value, an average or a mechanical formula, instead of as a slope at one point. Four mistakes account for most lost marks at this stage.

Mistake 1: Using An Average Rate When The Question Asks For An Instant

Where it slips in: "How fast is the stone falling at $t = 4$ s?" A student divides total distance by total time.

Don't do this: $\frac{78.4}{4} = 19.6,\text{m/s}$ is the average speed over 4 seconds, not the speed at $t = 4$.

The correct way: Differentiate first, then substitute: $d'(4) = 9.8(4) = 39.2,\text{m/s}$. The words "at", "when" and "instant" point to the derivative. "Over" and "between" point to an average.

Mistake 2: Applying The Power Rule To Something That Is Not A Power Of X

Where it slips in: $f(x) = 2^x$, where the variable sits in the exponent.

Don't do this: $f'(x) = x \cdot 2^{x-1}$. The power rule needs a fixed exponent on a variable base, and here the base is fixed.

The correct way: Exponential functions have their own rule: $\frac{d}{dx}\big[2^x\big] = 2^x \ln 2$. Check the shape (variable base, or variable exponent?) before choosing a rule.

The rusher is the student most likely to make this one. They see an exponent, reach for $n x^{n-1}$ and move on. The fix is a two-second habit: say out loud which part is the variable before writing anything.

Mistake 3: Differentiating A Product Factor By Factor

Where it slips in: Any product, such as $x \cdot x$ or $x^2 \sin x$.

Don't do this: Write the derivative of $x \cdot x$ as $1 \cdot 1 = 1$.

The correct way: Since $x \cdot x = x^2$, the true derivative is $2x$, not $1$. Products need the product rule, and quotients need the quotient rule. Neither works factor by factor.

Mistake 4: Mixing Up The Function Value And The Slope

Where it slips in: "Find the slope of $f(x) = x^2$ at $x = 3$." A student answers $9$.

Don't do this: $f(3) = 9$ is the height of the graph, not its steepness.

The correct way: $f'(3) = 2(3) = 6$. Height and slope are different questions about the same point.

The memorizer tends to fall here. They know the rules perfectly but have no picture of what $f'(a)$ means, so the height and the slope blur together. Students who sketch the tangent line once, even roughly, stop making this mistake almost immediately, because the drawing shows the slope is not the height. (It is also the distinction that is hardest to explain in words alone; the sketch does more than any rule.)

The real-world version of Mistake 1 is enforced on roads. A spot speed camera measures speed at one instant, close to a derivative.

An average speed camera times a car over a known distance and computes the average, so a driver who brakes for the camera and speeds between them is still caught. Scotland's A9 is a well-known example. The two readings answer different questions, just as $19.6$ and $39.2,\text{m/s}$ did for the stone.

Practice Problems On Introduction To Derivatives

Work each one before reading the answer. Every answer has been checked.

1. Differentiate $f(x) = x^4 - 3x^2 + 2x$ and find $f'(1)$. Answer to Question 1: $f'(x) = 4x^3 - 6x + 2$, and $f'(1) = 0$.

2. Find $f'(x)$ for $f(x) = \frac{6}{x^2}$ and evaluate it at $x = 1$. Answer to Question 2: $f'(x) = -\frac{12}{x^3}$, and $f'(1) = -12$.

3. Use the limit definition to find $f'(1)$ for $f(x) = 2x^2 + 1$. Answer to Question 3: the difference quotient simplifies to $4 + 2h$, so $f'(1) = 4$.

4. A ball's position is $s(t) = 5t^2 + 3t$ metres. Find its velocity at $t = 2$ s and its average velocity over $0$ to $2$ s. Answer to Question 4: $s'(2) = 23,\text{m/s}$, and the average is $13,\text{m/s}$.

5. Find every $x$ where the tangent to $f(x) = x^3 - 12x$ is horizontal, and the points on the curve there. Answer to Question 5: $x = -2$ and $x = 2$, at the points $(-2, 16)$ and $(2, -16)$.

6. Find the slope of $f(x) = x^{3/2}$ at $x = 4$. Answer to Question 6: $f'(x) = \frac{3}{2}x^{1/2}$, so $f'(4) = 3$.

Key Takeaways

• An introduction to derivatives starts with one idea: the derivative is the slope of the tangent line and the instantaneous rate of change at a point.

• The limit $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ turns shrinking average rates into one exact rate.

• The constant, power, constant multiple and sum rules differentiate any polynomial in one line.

• A derivative fails to exist at corners, vertical tangents and breaks in the graph.

• The most common mistakes are answering with an average or with the function value when the question asks for the slope at an instant.

Your Next Move

Work through Practice Problems 1 to 6. If Problem 3 trips you up, return to Example 2 and cancel the $h$ before letting it shrink. If Problem 4 does, reread "How Does Average Rate Of Change Become Instantaneous?".

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Read More

• Table of derivatives for common functions

• Applications of derivatives in maxima, minima and rates

• The second derivative and concavity

• Limits and derivatives together

• History of calculus from Newton and Leibniz onward