Finding the equation of a tangent and a normal

A tangent is the straight line that just touches a curve at a single point and runs in the same direction as the curve there. A normal is the straight line through that same point but at right angles to the tangent.

This method turns a calculus idea, the gradient of a curve, into the ordinary straight-line equation $y=mx+c$.

In Add Math this appears throughout the Differentiation chapter. A typical question gives you a curve and a point on it, then asks for the equation of the tangent, the normal, or both.

The key insight is that the gradient of the curve at a point, $\frac{dy}{dx}$ evaluated there, is exactly the gradient of the tangent line. Once you have a gradient and a point, the rest is coordinate geometry you already know.

Reach for this method whenever a question names a curve and a specific point and asks for the equation of a line touching or crossing it there. Trigger words include tangent, normal, 'the line that touches the curve', or 'the line perpendicular to the curve at'.

You will also need it when a question gives the gradient of the tangent instead of the point, for example, 'find the point where the tangent is parallel to $y=2x$'. There you set $\frac{dy}{dx}$ equal to the given gradient and solve for $x$ first.

Any time the words tangent or normal appear next to a curve, differentiation is your starting move: the derivative supplies the gradient, and a gradient plus a point always produces a line.

1. 1

   Differentiate

   Find $\frac{dy}{dx}$ for the curve.

2. 2

   Find the point

   If only the $x$-value is given, substitute it into the curve to find $y$, giving the point $(x_{1},y_{1})$.

3. 3

   Tangent gradient

   Substitute $x_{1}$ into $\frac{dy}{dx}$ to get the gradient $m$ of the tangent.

4. 4

   Normal gradient

   Take the negative reciprocal: $-\frac{1}{m}$. (If $m=0$ the normal is vertical.)

5. 5

   Write the equation

   Use $y-y_{1}=m(x-x_{1})$ with the correct gradient for the line you need.

6. 6

   Simplify

   Rearrange into $y=mx+c$ or $ax+by+c=0$ as asked.

• Using the $y$-value as the gradient. The gradient of the tangent is $\frac{dy}{dx}$ evaluated at the point, not the $y$-coordinate.
• Forgetting to find the $y$-coordinate when only $x$ is given, then having no point to substitute into $y-y_{1}=m(x-x_{1})$.
• Getting the normal gradient wrong, it is $-\frac{1}{m}$, the negative reciprocal, not $-m$ and not $\frac{1}{m}$.
• Substituting the $x$-value into $\frac{dy}{dx}$ before differentiating, or differentiating after substituting a number.
• Careless sign or fraction errors when clearing the $-\frac{1}{2}$ in the normal equation.

The step students most often trip on is the normal gradient: they either forget the negative reciprocal or mix up which line is which. In a one-to-one lesson our teachers watch you write the gradients side by side, so the pair $m$ and $-\frac{1}{m}$ becomes second nature and the two equations never get swapped.

Because Add Math is marked analytically, showing the derivative, the point and the substitution earns method marks even when a final sign slips. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

To see how we set out tangents and normals cleanly, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.