Revise Introduction to differentiation and rates of change for IB Mathematics: Applications and Interpretation (5.1)

Start with a quantity that changes, say the height of a growing plant. Measuring its height at the start and end of a week gives its average rate of growth — the gradient of the straight line (the secant) joining the two points. But the plant grew faster on sunny days and slower on cloudy ones, so to know its growth rate at one exact moment we let the two points slide together until the secant becomes a tangent touching the curve at a single point. The gradient of that tangent is the derivative: the instantaneous rate of change. These are two descriptions of the same number — the derivative is simultaneously the gradient of the curve and the rate of change of the quantity.

Average rate of change over $[x_1,x_2]$ (gradient of secant): $\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$ Instantaneous rate of change at $x=a$ (gradient of tangent): $f'(a)$

We write the derivative of $y=f(x)$ in two interchangeable notations. Lagrange's notation $f'(x)$ (say 'f-prime of x') is compact and good for evaluating at a point, as in $f'(2)$. Leibniz's notation $\frac{dy}{dx}$ (say 'dee-y by dee-x') makes the units obvious, because it literally reads as a change in $y$ over a change in $x$.

For polynomials and simple powers there is a fast, exact rule. To differentiate a single term, multiply by the current power and then reduce the power by one.

The power rule: $\dfrac{d}{dx}\left(ax^n\right) = a\,n\,x^{\,n-1}$

The derivative $f'(x)$ is itself a function — a formula for the gradient at any $x$. To get the gradient at one specific point, substitute that $x$-value into $f'(x)$. The result, $f'(a)$, is a single number: the gradient of the curve, and the instantaneous rate of change, at $x=a$.

The reason the derivative matters is that it measures a real rate of change. When a quantity is modelled by a function, its derivative tells you how fast that quantity is changing — and the units come straight from the model: (units of $y$) per (unit of $x$). A full interpretation always states three things: the value, whether the quantity is increasing (positive derivative) or decreasing (negative derivative), and the units.

In the exam the marks are analytic: each one is tied to a specific line of working. A method mark (M) rewards a correct approach — here, applying the power rule and substituting — even if the arithmetic later slips. An accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned first. Follow-through (FT) means that if you differentiate wrongly but then correctly substitute your own derivative, you still earn the substitution and final marks on your figure. All of that protection exists only if the method is written on the page. Study how each mark below is earned by a specific line.

Everything here is one skill — differentiate with the power rule, then read the result as a gradient or a rate of change. That skill is the gateway to the rest of calculus: setting $f'(x)=0$ to find the stationary points of a curve (maxima and minima), using the derivative to build the equation of a tangent line, and later reversing the process through integration to recover a quantity from its rate of change. Master the habit — rewrite as powers, differentiate term by term, substitute, then interpret with units — and the calculus that follows becomes variations on a method you already own.