Derivative of a Function: Definition, Rules & Examples

What Is The Derivative Of A Function?

The derivative of a function is its instantaneous rate of change: how fast the output climbs or falls as the input moves. For a function $f$, its derivative at a point $x$ is written $f'(x)$, and it answers one question, "if $x$ nudges forward by a hair, how fast does $f(x)$ respond right now?"

Two pictures describe the same number, and keeping both in view is the whole idea:

• Algebraically, the derivative is a limit of average rates of change. Over a step of width $h$, the average rate is $\dfrac{f(x+h) - f(x)}{h}$. Shrink $h$ toward zero and that average tightens onto a single instantaneous rate.

• Geometrically, the derivative is the slope of the tangent line to the graph of $f$ at the point $\big(x, f(x)\big)$. The tangent is the straight line that grazes the curve at that one point, and its steepness is $f'(x)$.

The formal statement pins this down with a limit:

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

The fraction inside is the difference quotient, the slope of the line through two nearby points on the curve. As $h \to 0$ those two points slide together, the connecting line pivots into the tangent, and the average rate becomes the instantaneous rate. This is the limit definition of derivative, and it is where every rule below comes from. For the full first-principles treatment, see the definition of the derivative.

The process of finding a derivative is called differentiation, and a function that has a derivative at $x$ is said to be differentiable at $x$.

How Do You Find A Derivative From The Definition?

Working from the limit shows why the rules are true. Take $f(x) = x^2$ and compute its derivative directly.

Start from the definition and expand $f(x+h) = (x+h)^2$:

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

The $x^2$ terms cancel, leaving a factor of $h$ that divides out:

$$f'(x) = \lim_{h \to 0} \frac{2xh + h^2}{h} = \lim_{h \to 0} \big(2x + h\big)$$

Now let $h \to 0$, and the stray $h$ vanishes:

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

Final answer: the derivative of $x^2$ is $2x$.

Read it geometrically. At $x = 3$ the slope is $2(3) = 6$; at $x = 0$ the slope is $0$, which is exactly the flat bottom of the parabola. The single formula $f'(x) = 2x$ hands you the tangent slope at every point at once.

What Notation Is Used For The Derivative Of A Function?

Several notations survive from different inventors, and you will meet all of them. They mean the same thing.

• Lagrange (prime): $f'(x)$, read "f prime of x." This is the working notation for a named function throughout this article.

• Leibniz: $\dfrac{dy}{dx}$, read "dee y by dee x," for $y = f(x)$. It keeps the "change in $y$ over change in $x$" idea visible and makes the chain rule read cleanly.

• Operator: $Df$ or $\dfrac{d}{dx}f(x)$, where $\dfrac{d}{dx}$ is treated as an instruction, "differentiate what follows."

• Newton (dot): $\dot{y}$, used mostly in physics for a derivative with respect to time.

To evaluate a derivative at a specific input, say $x = 2$, write $f'(2)$ or $\left.\dfrac{dy}{dx}\right|_{x=2}$. The choice of symbol is a matter of convenience, never of meaning.

What Are The Rules For Differentiating A Function?

Computing every derivative from the limit would be slow, so a handful of rules do the work. Each one is proved once from the definition, then reused forever. Below, $f$ and $g$ are differentiable functions and $c$ is a constant.

Table: The core differentiation toolbox, with one worked line for each rule.

Each rule earns its own worked example next. For the collected statements and proofs in one place, see rules of differentiation, and for the ready-made derivatives of standard functions, the table of derivatives.

How Does The Power Rule Work?

The power rule says the derivative of $x^n$ is $n,x^{n-1}$: drop the exponent in front, then reduce the exponent by one. With the constant-multiple rule alongside it:

$$\frac{d}{dx}\left(3x^4\right) = 3 \cdot 4x^{3} = 12x^{3}$$

This single rule, combined with the sum rule, differentiates every polynomial.

How Does The Product Rule Work?

The product rule handles a function multiplied by another function. It is not the product of the two derivatives. Take $f(x) = x^2\sin x$, with $u = x^2$ (so $u' = 2x$) and $v = \sin x$ (so $v' = \cos x$):

$$\frac{d}{dx}\left(x^2\sin x\right) = (2x)(\sin x) + (x^2)(\cos x) = 2x\sin x + x^2\cos x$$

Differentiate one factor, keep the other, and add the two crossed terms.

How Does The Quotient Rule Work?

The quotient rule differentiates one function divided by another. For $\dfrac{x}{x+1}$, take $f = x$ (so $f' = 1$) and $g = x+1$ (so $g' = 1$):

$$\frac{d}{dx}\left(\frac{x}{x+1}\right) = \frac{(1)(x+1) - (x)(1)}{(x+1)^2} = \frac{x + 1 - x}{(x+1)^2} = \frac{1}{(x+1)^2}$$

The order in the numerator matters: it is "bottom times derivative of top, minus top times derivative of bottom," all over the bottom squared.

How Does The Chain Rule Work?

The chain rule differentiates a composition, an outer function wrapped around an inner one. For $(3x+1)^5$, the outer function is "raise to the fifth" and the inner function is $g(x) = 3x+1$, with $g'(x) = 3$:

$$\frac{d}{dx}\left((3x+1)^5\right) = 5(3x+1)^{4} \cdot 3 = 15(3x+1)^4$$

Differentiate the outer layer, keep the inner layer untouched inside it, then multiply by the derivative of the inner layer. Forgetting that final inner factor is the most common slip in all of calculus, and it has its own entry in the mistakes section below.

What Are Higher-Order Derivatives Of A Function?

A derivative is itself a function, so you can differentiate it again. The result is the second derivative, written $f''(x)$, and repeating gives third, fourth, and higher orders.

Take $f(x) = x^3$ and differentiate repeatedly:

$$f'(x) = 3x^2, \qquad f''(x) = 6x, \qquad f'''(x) = 6, \qquad f^{(4)}(x) = 0$$

Each order has a plain meaning. The first derivative is the rate of change; the second derivative is the rate at which that rate itself changes. In motion, if $f$ is position, then $f'$ is velocity and $f''$ is acceleration, the everyday reason the second derivative matters.

What Does The Derivative Of A Function Tell You?

Because the derivative is the slope of the graph, its sign and its zeros read off the shape of the curve at a glance.

• Increasing or decreasing. Where $f'(x) > 0$ the function is rising; where $f'(x) < 0$ it is falling. The sign of the slope is the direction of the curve.

• Turning points. Where $f'(x) = 0$ the tangent is flat, marking a possible maximum, minimum, or a level pause. These are the critical points a curve turns at.

• Concavity. The second derivative sets the bend: $f''(x) > 0$ curves the graph upward like a valley, and $f''(x) < 0$ curves it downward like a hill.

These three readings are the engine behind curve sketching and optimisation, gathered under applications of derivatives. The tangent line itself, the geometric heart of the derivative, is built at tangent line equations.

Does Differentiable Mean Continuous?

Yes in one direction, and the direction matters. If a function is differentiable at a point, then it is continuous there. A curve can only have a well-defined tangent slope where it has no break or jump, so differentiability is the stronger condition.

The converse is false. A function can be continuous yet fail to be differentiable, and the standard example is $f(x) = |x|$ at $x = 0$: the graph is unbroken, but it has a sharp corner, and the slope coming from the left ($-1$) disagrees with the slope from the right ($+1$). With no single tangent, there is no derivative there. Continuity is necessary for differentiability, but not enough on its own.

Why Does The Derivative Of A Function Work?

The derivative feels almost too powerful the first time, one formula giving the exact slope everywhere. The reason it holds together is the limit that defines it.

• Averages sharpen into an instant. You cannot measure a rate at a single frozen instant by direct division, since that is $0/0$. The limit sidesteps this: it watches the average rate over shorter and shorter intervals and reports the value they close in on.

• The tangent is the best straight-line fit. Near the point of contact, the tangent line hugs the curve more tightly than any other line. Its slope is the derivative because it captures the curve's direction at that exact spot.

• Local, not global. A derivative describes behaviour right at a point, not over a stretch. That locality is what lets it track a curve whose steepness changes from place to place.

Seen this way, differentiation is just careful zooming. Zoom far enough into a smooth curve at one point and it looks straight, and the slope of that straight-line view is the derivative.

Who Discovered The Derivative Of A Function?

The idea of a tangent slope was chased for decades before it had a name or a symbol.

Two more names shaped how we write and ground the derivative today:

• Joseph-Louis Lagrange (1736–1813, Italy/France) introduced the prime notation $f'(x)$ in 1797, the compact symbol used throughout this article.

• Augustin-Louis Cauchy (1789–1857, France) put the derivative on the rigorous limit footing every modern course uses, defining it exactly as the limit of the difference quotient.

Where Is The Derivative Of A Function Used In The Real World?

Any time a quantity changes and you want its rate, you are taking a derivative, whether or not anyone writes $f'(x)$.

• Physics and motion: velocity is the derivative of position, and acceleration is the derivative of velocity, so every speedometer and every crash-test model runs on derivatives.

• Economics: marginal cost and marginal revenue are the derivatives of total cost and total revenue, telling a firm what one more unit adds.

• Biology and medicine: the growth rate of a population, a tumour, or a drug's concentration in the blood is the derivative of the amount with respect to time.

• Engineering and control: rates of temperature, pressure, and current drive the feedback systems that keep aircraft, reactors, and thermostats stable.

• Machine learning: training a model means following derivatives (gradients) of an error function downhill to reduce the error step by step.

One idea, the rate of change of a quantity, connects a falling apple, a company's pricing, and a self-driving car's steering. That reach is why the derivative is the first big tool of calculus.

What Are The Most Common Mistakes With The Derivative Of A Function?

These three errors account for most lost marks on differentiation, and each matches a question students actually ask on r/calculus, r/learnmath, and course common-error handouts from MIT and Georgia Tech.

Using the power rule on an exponential function.

Where it slips in:

A student sees $2^x$ or $e^x$ and, on autopilot, "brings the power down," writing $\frac{d}{dx}\left(2^x\right) = x,2^{x-1}$.

Don't do this:

Do not apply the power rule when the variable is in the exponent. The power rule $\frac{d}{dx}(x^n) = n x^{n-1}$ is only for a variable base raised to a constant power.

The correct way:

Use the exponential rule: $\frac{d}{dx}\left(e^x\right) = e^x$, and $\frac{d}{dx}\left(a^x\right) = a^x \ln a$. The base $x^2$ and the exponential $2^x$ are opposite cases and differentiate by opposite rules.

Treating the derivative of a product as the product of the derivatives.

Where it slips in:

For $x^2\sin x$, a student writes $\frac{d}{dx}\left(x^2\sin x\right) = (2x)(\cos x)$, differentiating each factor and multiplying.

Don't do this:

Do not multiply the two derivatives together. In general $\big(fg\big)' \neq f'g'$.

The correct way:

Apply the product rule: keep one factor while differentiating the other, then add. $\frac{d}{dx}\left(x^2\sin x\right) = 2x\sin x + x^2\cos x$.

Forgetting the inner factor in the chain rule.

Where it slips in:

For $(3x+1)^5$, a student writes $5(3x+1)^4$ and stops, treating the inside as if it were a bare $x$.

Don't do this:

Do not skip the derivative of the inner function. Failing to apply the chain rule is the single most common error in differential calculus.

The correct way:

Multiply by the derivative of the inside. Since the inner function $3x+1$ has derivative $3$, the answer is $5(3x+1)^4 \cdot 3 = 15(3x+1)^4$.

Practice Problems On The Derivative Of A Function

Work each one, then check against the answer. Every answer is verified.

1. Differentiate $f(x) = 5x^3$.
   (Answer: $f'(x) = 15x^2$, by the power and constant-multiple rules.)

2. Differentiate $f(x) = x\cos x$.
   (Answer: $f'(x) = \cos x - x\sin x$, by the product rule.)

3. Differentiate $f(x) = \dfrac{x^2}{x-1}$.
   (Answer: $f'(x) = \dfrac{x(x-2)}{(x-1)^2}$, by the quotient rule.)

4. Differentiate $f(x) = (2x^2 + 1)^4$.
   (Answer: $f'(x) = 16x(2x^2+1)^3$, by the chain rule.)

5. From the definition, differentiate $f(x) = 3x + 2$.
   (Answer: $f'(x) = 3$, since $\frac{3(x+h)+2 - (3x+2)}{h} = 3$ for all $h$.)

6. Find the second derivative of $f(x) = x^4$.
   (Answer: $f'(x) = 4x^3$, so $f''(x) = 12x^2$.)

Where Should You Go Next After The Derivative Of A Function?

The derivative is the hub of differential calculus, and several natural doors open from here.

1. Definition of the derivative. Go deeper on the limit and the difference quotient that every rule is built from.

2. Rules of differentiation. Master the power rule, product rule, quotient rule, and chain rule as a system.

3. Applications of derivatives. Use the derivative to find maxima and minima, sketch curves, and write tangent line equations.

If your child is building these foundations, a live Bhanzu trainer teaches the derivative from the tangent-slope picture up in the Bhanzu math program.