What does dy/dx give at a point on a curve?

the gradient of the curve. It is also the gradient of the tangent there.

The tangent to a curve at a point has gradient 4. What is the gradient of the normal?

−1/4. The negative reciprocal.

The tangent to a curve at (1, 5) has gradient 2. What is its equation?

y = 2x + 3. y − 5 = 2(x − 1).

Gradients, tangents and normals · Cambridge IGCSE Additional Mathematics

In plain words

The derivative at a point is the gradient of the curve there. A tangent is the straight line that just touches the curve at that point, with that gradient. A normal is the line through the same point at right angles to it.

5 things to know

The gradient of a curve at a point is the value of dy/dx there.
The tangent at a point has the same gradient as the curve at that point.
The normal is perpendicular to the tangent. If the tangent has gradient m, the normal has gradient −1/m.
To find either line: find the y-coordinate of the point, find the gradient from dy/dx, then use y − y₁ = m(x − x₁).
To find where a curve has a given gradient, put dy/dx equal to that gradient and solve.

Worked example

Find the equations of the tangent and the normal to y = x³ − 2x at the point where x = 2.

When x = 2, y = 8 − 4 = 4. The point is (2, 4).
dy/dx = 3x² − 2. When x = 2, the gradient is 10.
Tangent: y − 4 = 10(x − 2), which is y = 10x − 16.
Normal: gradient −1/10. y − 4 = −(1/10)(x − 2), which is x + 10y = 42.

Tips and tricks

Substitute the x-value into dy/dx to get a number before writing the equation of the line. A gradient with x still in it is not a gradient.
Tangent: same gradient. Normal: negative reciprocal. Read the question for which is wanted.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Gradients, tangents and normals: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What does dy/dx give at a point on a curve?
- the y-coordinate
- the gradient of the curve (the answer)
- the area under the curve
- the normal
It is also the gradient of the tangent there.
The tangent to a curve at a point has gradient 4. What is the gradient of the normal?
- 4
- −4
- 1/4
- −1/4 (the answer)
The negative reciprocal.
What is the gradient of y = x² at x = 3?
- 3
- 6 (the answer)
- 9
- 2
dy/dx = 2x.
What is the gradient of y = x³ at x = 1?
- 1
- 3 (the answer)
- 0
- 2
dy/dx = 3x².
The tangent to a curve at (1, 5) has gradient 2. What is its equation?
- y = 2x + 5
- y = 2x + 3 (the answer)
- y = 5x + 2
- y = 2x − 3
y − 5 = 2(x − 1).

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 10 marks. Write your answers on paper, then check them.

Gradients, tangents and normals

Cambridge IGCSE Additional Mathematics 0606 · 10 marks · papermunch.org

Name ______________________________ Date ______________

Find the equation of the tangent to y = x² + 3x at the point where x = 1.[3]
Show answerHide answer
y = 5x − 1. The point is (1, 4) and the gradient is 5.
I got this one
Find the equation of the normal to y = 2x² − 5 at the point (2, 3).[4]
Show answerHide answer
x + 8y = 26. The tangent's gradient is 8, so the normal's is −1/8.
I got this one
Find the x-coordinates of the points on y = x³ − 12x where the gradient is zero.[3]
Show answerHide answer
x = 2 and x = −2. 3x² − 12 = 0.
I got this one

Answers: Gradients, tangents and normals

1. y = 5x − 1. The point is (1, 4) and the gradient is 5.
2. x + 8y = 26. The tangent's gradient is 8, so the normal's is −1/8.
3. x = 2 and x = −2. 3x² − 12 = 0.