Sign Errors in the Normal

The error

The gradient of the normal is the negative reciprocal of the gradient of the tangent: $m_{\text{normal}}=-\frac{1}{m_{\text{tangent}}}$ . The trouble starts when the tangent gradient is itself negative.

Now the formula contains two minus signs, the one in front of the fraction and the one inside $m_{\text{tangent}}$ , and they cancel to give a positive normal gradient. Under exam pressure students write the formula correctly but then let one minus sign quietly vanish, so a normal that should have gradient $+\frac{1}{2}$ is recorded as $-\frac{1}{2}$ .

It is tempting because writing a single leading minus feels like 'the negative reciprocal', and the double negative is easy to overlook. From there the whole normal line is wrong, even though the tangent gradient was found correctly.

Wrong line, then the fix

For the curve $y=x^{2}-4x+5$ at the point where $x=1$ , the tangent gradient is $\frac{dy}{dx}=2x-4=2(1)-4=-2$ . The negative reciprocal is then formed.

The tempting slip loses one of the two minus signs:

Wrong, the double negative was droppedMust memorise

m_{\text{normal}}=-\frac{1}{-2}=-\frac{1}{2}

Handled correctly, the minus in front and the minus inside cancel:

CorrectMust memorise

m_{\text{normal}}=-\frac{1}{-2}=+\frac{1}{2}

A quick sanity check settles it: the tangent slopes down (gradient $-2$ ), so a line at right angles to it must slope up, and an upward slope is positive. The normal gradient has to be $+\frac{1}{2}$ .

The reliable fix

1
Find the tangent gradient
Differentiate and evaluate $\frac{dy}{dx}$ at the given point; call this $m_{T}$ .
2
Write the formula in full
Write $m_{N}=-\dfrac{1}{m_{T}}$ before substituting, keeping every sign visible.
3
Substitute inside brackets
If $m_{T}$ is negative, put it in brackets: $-\dfrac{1}{(-2)}$ , so the two minus signs are impossible to miss.
4
Simplify the double sign
Two minus signs give a plus; a negative tangent gradient always produces a positive normal gradient, and vice versa.
5
Sanity-check the direction
Tangent and normal gradients must have opposite signs (unless one line is horizontal or vertical).

One line to remember it

Flip it, then switch the sign

Tangent and normal gradients always have opposite signs. If your tangent gradient was negative and your normal gradient came out negative too, a minus sign went missing.

Practice where the error hides

Q1[4 marks]

The curve $y=x^{3}-6x^{2}+9x$ passes through the point $(2,2)$ . Find the equation of the normal to the curve at this point.

Show worked solution

First differentiate to get the tangent gradient.

\frac{dy}{dx}=3x^{2}-12x+9

At $x=2$ : $3(2)^{2}-12(2)+9=12-24+9=-3$ , so $m_{T}=-3$ .

The normal gradient is the negative reciprocal. Keep the negative tangent gradient in brackets so both minus signs stay in view:

m_{N}=-\frac{1}{(-3)}=+\frac{1}{3}

The tangent slopes down and the normal slopes up, so a positive $\frac{1}{3}$ is correct. Use the point $(2,2)$ in $y-y_{1}=m_{N}(x-x_{1})$ :

y-2=\frac{1}{3}(x-2)

Multiply through by $3$ : $3y-6=x-2$ , which rearranges to the equation of the normal:

x-3y+4=0

Check: at $(2,2)$ , $2-3(2)+4=2-6+4=0$ , so the point lies on the line.

How one-to-one teaching helps

The sign slip is not a knowledge gap, students know the formula, so our teachers fix it as a habit, not a lecture. We insist on two things every time: write $m_{N}=-\frac{1}{m_{T}}$ in full before substituting, and put any negative gradient in brackets so the double negative cannot hide.

Then we add a one-second direction check: down-sloping tangent, up-sloping normal. In a one-to-one lesson we can catch the exact moment the sign disappears and rebuild the reflex.

Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English. Message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.

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Frequently asked questions

Why is the normal gradient the negative reciprocal?

The normal is perpendicular to the tangent, and two perpendicular lines (neither horizontal nor vertical) have gradients whose product is $-1$ . So if the tangent gradient is $m_{T}$ , the normal gradient must satisfy $m_{T}\times m_{N}=-1$ , which gives $m_{N}=-\frac{1}{m_{T}}$ .

What happens to the sign when the tangent gradient is negative?

The normal gradient becomes positive. The minus in front of the fraction and the minus inside the negative gradient cancel, for example $-\frac{1}{-2}=+\frac{1}{2}$ .

Tangent and normal gradients always end up with opposite signs.

How can I check my normal gradient quickly?

Multiply it by the tangent gradient, the product should be exactly $-1$ . If you get $+1$ , a sign is wrong.

A direction check also works: a downhill tangent needs an uphill (positive) normal.

Do I lose all the marks for a sign error?

Not necessarily. Under analytic marking you keep method marks for the correct tangent gradient and the correct use of $y-y_{1}=m(x-x_{1})$ , but you lose the accuracy of the final answer.

Writing the negative reciprocal step clearly makes the intended method easy to credit.

Source:SRC-DSKP-EN

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026

Sign errors when finding the normal

The error

Wrong line, then the fix

The reliable fix

One line to remember it

Practice where the error hides

How one-to-one teaching helps

Frequently asked questions

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