Definition of Gradient (Cambridge International AS Mathematics)

Definition of gradient

What is the gradient of a curve?

• At a given point the gradient of a curve is defined as the gradient of the tangent to the curve at that point
• A tangent to a curve is a line that just touches the curve at one point but generally doesn't cut the curve at that point

• A tangent may cut the curve somewhere else on the curve

• At a point of inflection the tangent may cross the curve

  • E.g. The tangent to $y=x^{3}$ at $x=0$

• It is only possible to draw one tangent to a curve at any given point

• Note that unlike the gradient of a straight line, the gradient of a curve is constantly changing

Exam Hint

• A tangent only exists at points where a curve is smooth.
• For example, there is no tangent (or gradient) at one of the 'corners' in a modulus function.

Worked Example