If you differentiate the equation of a curve, you will get a formula for the gradient of the curve. Before you learnt calculus, you would have found the gradient of a curve by drawing a tangent to the curve and measuring the gradient of this. This is because the gradient of a curve at a point is equal to the gradient of the tangent at that point.
Find the equation of the tangent to the curve y = x³ at the point (2, 8).
dy = 3x²
Gradient of tangent when x = 2 is 3×2² = 12.
From the coordinate geometry section, the equation of the tangent is therefore:
y - 8 = 12(x - 2)
so y = 12x - 16
You may also be asked to find the gradient of the normal to the curve. The normal to the curve is the line perpendicular (at right angles) to the tangent to the curve at that point.
Remember, if two lines are perpendicular, the product of their gradients is -1.
So if the gradient of the tangent at the point (2, 8) of the curve y = x³ is 8, the gradient of the normal is -1/8, since -1/8 × 8 = -1.