14.5 Gradients, Tangents and Normals

2026 📋 Syllabus Objectives

By the end of these notes, you should be able to:

  • Use differentiation to find the gradient of a curve at any point
  • Find the equation of a tangent to a curve at a given point
  • Find the equation of a normal to a curve at a given point

📖 Main Content

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1. What is a Gradient?

The gradient of a curve at any point tells you how steep the curve is at that exact point. Unlike a straight line (which has the same steepness everywhere), a curve's steepness changes from point to point.

To find the gradient of a curve at a specific point, you:

  1. Differentiate the equation of the curve to get dydx\frac{dy}{dx}
  2. Substitute the x-value of the point into dydx\frac{dy}{dx}
  3. The result is the gradient, mm, at that point

💡 Remember: dydx\frac{dy}{dx} is the derivative — it gives you a formula for the gradient at any x-value.

2. What is a Tangent?

A tangent is a straight line that just touches a curve at one point without crossing through it. At the point of contact, the tangent has exactly the same gradient (steepness) as the curve.

Think of it like this: if you zoomed in very closely on the curve at a point, it would look like a straight line — that straight line is the tangent.

How to Find the Equation of a Tangent

To find the equation of a tangent at a point (x1,y1)(x_1, y_1):

Step 1: Differentiate yy to get dydx\frac{dy}{dx}.

Step 2: Substitute x=x1x = x_1 into dydx\frac{dy}{dx} to find the gradient mm.

Step 3: If you are only given the x-value, substitute it into the original equation y=...y = ... to find y1y_1.

Step 4: Use the straight-line formula:

y−y1=m(x−x1)\boxed{y - y_1 = m(x - x_1)}

where (x1,y1)(x_1, y_1) is the point on the curve and mm is the gradient found in Step 2.

3. What is a Normal?

A normal is a straight line that is perpendicular (at a right angle, 90°) to the tangent at the same point on the curve.

The Key Relationship Between Tangent and Normal Gradients

If two lines are perpendicular to each other, their gradients multiply to give −1-1. This means:

mtangent×mnormal=−1m_{\text{tangent}} \times m_{\text{normal}} = -1

So, if the gradient of the tangent is mm, then:

mnormal=−1mm_{\text{normal}} = -\frac{1}{m}

In plain English: flip the tangent gradient and change its sign to get the normal gradient.

⚠️ This only works when m≠0m \neq 0. If the tangent is horizontal (gradient = 0), the normal is a vertical line.

How to Find the Equation of a Normal

Step 1: Find the gradient of the tangent, mm, at the point (same as for the tangent — differentiate and substitute the x-value).

Step 2: Calculate the gradient of the normal: −1m-\frac{1}{m}.

Step 3: Use the same straight-line formula, but with the normal's gradient:

y−y1=−1m(x−x1)\boxed{y - y_1 = -\frac{1}{m}(x - x_1)}

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