Differentiation as a Rate of Change for the ESAT

What is a rate of change?

To understand the derivative, one must first master the concept of a rate of change. A common example is speed. Speed describes how quickly distance changes relative to time. For instance, a speed of $3$ m/s means distance increases by $3$ metres for every $1$ second that passes. Rates of change quantify how many units of one measure change per single unit of another. Acceleration is another example, representing the change in speed per unit of time. It is measured in metres per second changed for every second, often written as metres per second per second, or $m/s^2$ (alternatively $ms^{-2}$).

Gradients as rates of change

Gradients are fundamentally rates of change. For a straight line, the gradient indicates how much the vertical value $y$ changes for every $1$ unit move horizontally in the $x$ direction. This means the gradient is the rate of change of $y$ with respect to $x$. This principle applies to graphs in various contexts: the gradient of a distance-time graph represents speed, while the gradient of a speed-time graph represents acceleration.

The derivative of a curve at a point

While the rate of change for a straight line is constant, a curve has a changing rate 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 definition allows us to treat a complex curve as a series of local linear approximations. Differentiation is the mathematical process used to find this gradient.

Differentiation notation and meaning

It is essential to distinguish between a function and its derivative. If we have an expression such as $y = x^3 + 7x^2 - 3x + 11$, substituting a value for $x$ provides the corresponding $y$ coordinate on the curve. However, the derivative, denoted as $\frac{dy}{dx}$, provides the gradient (rate of change) at that point. For the expression above, the derivative is $\frac{dy}{dx} = 3x^2 + 14x - 3$. By substituting an $x$ value into this new expression, we find the gradient of the tangent to the curve at that specific $x$ coordinate.

Candidates must be familiar with several notations:

1. Leibniz notation: $\frac{dy}{dx}$ for the first derivative and $\frac{d^2y}{dx^2}$ for the second derivative.
2. Lagrange notation: $f'(x)$ for the first derivative and $f''(x)$ for the second derivative.
3. Dot notation: In physics or mechanics, differentiation with respect to time is sometimes written as $\dot{s}$ or $\ddot{s}$, though this is less common in the ESAT Mathematics papers.

Note the placement of the numerals in second-order notation: $\frac{d^2y}{dx^2}$. The $2$ is placed after the $d$ in the numerator and after the $x$ in the denominator.

Calculating the derivative

The ESAT focuses on the differentiation of simple expressions involving sums of powers of $x$. The fundamental rule for a power of $x$ is:

$\frac{d}{dx} x^n = nx^{n-1}$

In this context, the operator $\frac{d}{dx}$ means “differentiate this with respect to $x$”. When faced with complex looking expressions, the most effective strategy is to simplify them into a sum of powers of $x$ before differentiating. You can differentiate an expression term by term. For example:

$\frac{d}{dx}(x^3 + 7x^2 - 3x + 11) = \frac{d}{dx}x^3 + \frac{d}{dx}7x^2 - \frac{d}{dx}3x + \frac{d}{dx}11 = 3x^2 + 14x - 3 + 0$

Note that the derivative of a constant (like $11$) is always $0$, as a constant value does not change, meaning its rate of change is zero.

Second order derivatives

A second order derivative is simply the derivative of the first derivative. If $\frac{dy}{dx}$ tells us how $y$ changes with $x$, then $\frac{d^2y}{dx^2}$ tells us how the gradient itself is changing. This is particularly useful for identifying the nature of stationary points or understanding acceleration in kinematics.