Differentiation 6: Tangents and normals

Overview:
  1. Gradient of a line passing through two points
  2. Limits
  3. Differentiation from first principles
  4. Differentiating expressions of the form \(kx^n\) with respect to \(x\)
  5. The gradient at a point on a curve
  6. Tangents and normals

 Part 6: Tangents and normals


Skill recap: Finding the equation of a line through a point with a given gradient

You should know that the line with gradient \(m\) through the point with coordinates \(x_1,y_1)\) is \(y-y_1=m(x-x_1)\). You can practise using this formula using this applet:

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The equation of a line of the tangent or normal to a line at a given point

A tangent to a curve at a given point just touches the curve without crossing it at that point.

A normal or normal line to a curve at a given point is the line perpendicular to the tangent.

Recall that if two lines at right angles to each other have gradients \(m_1\) and \(m_2\), then \(m_1m_2 =-1\) which can be rearranged as \(m_1=\dfrac{-1}{m_2}\). In other words, the gradient of each line is the negative reciprocal of the other.

In the part 5, we saw how to calculate the gradient at a point on a curve. Using this knowledge together with the formula and facts above, we can find the equation of a tangent or a normal to a curve at a given point. In the applet below, use the toggle to change between finding a tangent and finding a normal. Drag down the slider for step-by-step solutions to each question, and generate new questions for more practice.

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