Revise Tangents, normals and the second derivative for IB Mathematics: Analysis and Approaches (5.3)

The gradient of a straight line is constant, but the gradient of a curve changes from point to point. The derivative $f'(x)$ is precisely the rule that returns the gradient at each $x$. So to find how steep the curve $y=f(x)$ is at the point where $x=a$, you differentiate once to get $f'(x)$ and then substitute: the gradient there is $f'(a)$. This single value is the gradient of the tangent line at that point — the straight line that just touches the curve and heads in the same direction as the curve at that instant.

A tangent to a curve at a point is the straight line through that point whose gradient equals the derivative there. Finding its equation is a fixed three-step routine, and the step students most often drop is the very first one — the $y$-coordinate of the point.

To find the tangent to $y=f(x)$ at the point where $x=a$:

1. Find $y_1 = f(a)$, so the point is $(a, f(a))$.
2. Differentiate and evaluate the gradient $m_T = f'(a)$.
3. Use the point–gradient form $y - f(a) = f'(a)(x - a)$.

The normal to a curve at a point is the line perpendicular to the tangent there. Because perpendicular lines have gradients whose product is $-1$, once you know the tangent gradient $m_T = f'(a)$ the normal gradient follows immediately as $m_N = -1/f'(a)$. The point is the same $(a, f(a))$; only the gradient changes.

The sign of the first derivative tells you the direction of the curve. Where $f'(x) > 0$ the gradient is positive and the function is increasing (rising left to right); where $f'(x) < 0$ the function is decreasing. To find the intervals, differentiate, then solve the inequality $f'(x) > 0$ or $f'(x) < 0$. The boundaries between increasing and decreasing behaviour are exactly the points where $f'(x) = 0$.

A stationary point is a point where $f'(x) = 0$: the tangent is momentarily horizontal. There are three kinds — a local maximum (the curve turns from rising to falling), a local minimum (falling to rising), and a horizontal point of inflection (the curve levels off but keeps going the same way). To find them, solve $f'(x)=0$; to classify them, examine how the curve behaves around each point using either the first- or the second-derivative test.

Differentiating the derivative gives the second derivative $f''(x)$, also written $\dfrac{d^2y}{dx^2}$. Where $f'$ describes the slope, $f''$ describes how the slope is changing — the concavity of the curve. If $f''(x) > 0$ the gradient is increasing and the curve is concave up (a cup); if $f''(x) < 0$ the gradient is decreasing and the curve is concave down (a frown). This gives the fast second-derivative test for a stationary point $x=a$: substitute $a$ into $f''$. A positive value means a minimum, a negative value means a maximum, and a zero value means the test is inconclusive — you must then fall back on the sign of $f'$ either side.

A point of inflection is where the concavity changes — from concave up to concave down, or vice versa. For this the second derivative must be zero AND change sign. Solving $f''(x)=0$ finds the candidate; you then confirm the sign of $f''$ actually flips across it. In the example above, $h''(x) = 12x - 18 = 0$ at $x = \tfrac{3}{2}$; for $x < 1.5$, $h''<0$ (concave down) and for $x > 1.5$, $h''>0$ (concave up), so the concavity changes and $x = 1.5$ is a point of inflection.

On Paper 1 the marks are analytic: each is tied to a specific line of working — a method mark (M) or an accuracy mark (A) — and the A-marks are dependent, meaning they are only awarded when the method they rest on is present. Follow-through (FT) means a correct method built on a wrong earlier value can still score the later marks, and 'in-symbolic-work' (ISW) means once you have a correct answer, a subsequent tidy-up slip is ignored. Equivalent forms are accepted. But that protection only exists if your method is written down. Study how each mark below is earned by a specific line.

Reading a curve through its derivatives is the engine behind optimisation, where you maximise or minimise a real quantity by finding and classifying a stationary point, and behind curve sketching, where increasing/decreasing intervals, turning points, concavity and inflections combine into a full picture of a graph. The same $f'$ and $f''$ reappear in kinematics as velocity and acceleration. Master the habit — point first, then gradient, then classify with $f''$, and always justify a sign change — and the applications that follow become variations on a method you already own.