Differentiation Derivatives and Rates of Change

Updated August 2025

This topic introduces differentiation as the tool for measuring how functions change. You will learn to interpret the derivative as the gradient of a tangent and a rate of change, and become familiar with the notation for both first and second order derivatives essential for the TMUA.

Core concept

The derivative of a function $f(x)$ at a point is the instantaneous rate of change of the function with respect to $x$ , which is geometrically represented as the gradient of the tangent to the graph $y = f(x)$ at that point.

Understanding Rates of Change

To understand what a derivative is, you must first have a firm grasp of the concept of a rate of change. A rate of change tells you how fast one variable changes in comparison to another. A common real world example is speed: speed is the rate of change of distance compared to time. A speed of $3$ m/s means that distance is changing at a rate of $3$ metres for every one second of time that passes.

Acceleration is another example: it is a rate of change that tells you how much speed changes for every unit of time. This is measured in metres per second changed per second, or $m/s^2$ .

In mathematics, gradients are also rates of change. The gradient of a straight line tells us how much we must move vertically to return to the line for every $1$ unit we move horizontally. In other words, the gradient is the rate of change of $y$ with respect to $x$ . On a distance-time graph, the gradient represents speed: on a speed-time graph, the gradient represents acceleration.

The Derivative as the Gradient of a Tangent

While the rate of change for a straight line is constant and easy to identify (it is simply the gradient), curves have changing rates of change. We define the rate of change at a specific point on a curve as the gradient of the tangent to the curve at that point.

This intuition is the foundation of calculus. If you consider an expression such as $y = f(x)$ , the function tells you the $y$ value for any given $x$ . The derivative, however, tells you the gradient of the curve (the rate of change) at that point. For example, if we have the expression $y = x^3 + 7x^2 - 3x + 11$ , we can find the $y$ value by substitution. The derivative of this expression is $\frac{dy}{dx} = 3x^2 + 14x - 3$ . This new expression allows us to calculate the gradient of the tangent to the curve for any given value of $x$ .

Notation for Derivatives

There are several standard notations used to represent derivatives in the TMUA. You must be comfortable with all of them:

Leibniz notation: $\frac{dy}{dx}$ represents the first derivative of $y$ with respect to $x$ .
Function notation: $f'(x)$ represents the derivative of the function $f(x)$ .

While not used directly in the TMUA, you may also see the dot notation in physics or mechanics, where $\dot{s}$ represents $\frac{ds}{dt}$ (differentiation with respect to time).

Second Order Derivatives

The derivative of an expression is itself a function of $x$ , which means it can be differentiated again. This is called the second-order derivative.

In Leibniz notation, this is written as $\frac{d^2y}{dx^2}$ . It is important to note exactly where the two $2$ s are placed in this symbol.
In function notation, this is written as $f''(x)$ .

Just as the first derivative represents the rate of change of the original function, the second derivative represents the rate of change of the gradient itself. This is highly useful when determining the nature of stationary points, as it tells us whether the gradient is increasing or decreasing at a specific point.

Key takeaways

The gradient of a curve at a point is defined as the gradient of the tangent at that point.
A derivative measures the rate of change of one variable with respect to another.
$\frac{dy}{dx}$ and $f'(x)$ are different notations for the same concept: the first derivative.
The second derivative, $\frac{d^2y}{dx^2}$ or $f''(x)$ , is the derivative of the first derivative.
Differentiation from first principles is not required for the TMUA, but the conceptual understanding of tangents is essential.

Tips

In TMUA questions, if you are asked about the 'rate of increase' of a quantity, you should immediately think of finding the derivative. Always check if a question is asking for the value of the function itself or the gradient of the function.

Cautions

Do not confuse the second derivative $\frac{d^2y}{dx^2}$ with the square of the first derivative $(\frac{dy}{dx})^2$ . They are entirely different mathematical operations.

Insight

The second derivative is to the first derivative what acceleration is to velocity. If the first derivative is the 'speed' of the $y$ value, the second derivative is the 'acceleration' of the $y$ value.

Worked Examples

Example 1

Which one of A-F correctly completes the following statement?
Given that

a < b

, and

f(x) > 0

for all

x

with

a < x < b

, the trapezium rule produces an overestimate for

\int_a^b f(x)\,dx

...

A:... if

f'(x) > 0

and

f''(x) < 0

for all

x

with

a < x < b

B:... only if

f'(x) > 0

and

f''(x) < 0

for all

x

with

a < x < b

C:... if and only if

f'(x) > 0

and

f''(x) < 0

for all

x

with

a < x < b

D:... if

f'(x) < 0

and

f''(x) > 0

for all

x

with

a < x < b

E:... only if

f'(x) < 0

and

f''(x) > 0

for all

x

with

a < x < b

F:... if and only if

f'(x) < 0

and

f''(x) > 0

for all

x

with

a < x < b

Practice Questions

Practice Question 1

f(x)

is a polynomial function defined for all real

x

.
Which of the following is a necessary condition for the inequality

\frac{f(a)+f(b)}{2} \ge f\left(\frac{a+b}{2}\right)

to be true for all real numbers

a

and

b

with

a < b

f(x) \ge 0

for all real

x

f'(x) \ge 0

for all real

x

f''(x) \ge 0

for all real

x

f(x) \le 0

for all real

x

f'(x) \le 0

for all real

x

f''(x) \le 0

for all real

x

Frequently asked questions

What is the difference between $y$ and $\frac{dy}{dx}$ ?

$y$ represents the position or value of the function on the vertical axis for a given $x$ , while $\frac{dy}{dx}$ represents the steepness or gradient of the function at that same $x$ value.

Do I need to know how to differentiate from first principles for the TMUA?

No, the official specification explicitly excludes differentiation from first principles, though you must understand the derivative's relationship to the tangent.

What does a negative derivative mean?

A negative derivative indicates that the $y$ values are decreasing as the $x$ values increase, meaning the graph is sloping downwards from left to right.

Where do the 2s go in the second derivative notation?

In Leibniz notation, the $2$ is placed after the $d$ in the numerator and after the $x$ in the denominator: $\frac{d^2y}{dx^2}$ .

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Solving Exponential Equations Logarithms

Differentiation Of Rational Powers

Understanding Rates of Change

The Derivative as the Gradient of a Tangent

Notation for Derivatives

Second Order Derivatives

Worked Examples

Practice Questions

Frequently asked questions

What is the difference between yyy and dydx\frac{dy}{dx}dxdy​?

Do I need to know how to differentiate from first principles for the TMUA?

What does a negative derivative mean?

Where do the 2s go in the second derivative notation?

Ready to test your knowledge?

What is the difference between $y$ and $\frac{dy}{dx}$ ?