Differentiation (revision)

Rule: Bring down the power and reduce the power by 1.

y =

14x + 12

Finding turning points (maximum/ minimum values) A turning point occurs where the gradient is zero, i.e. where 0.

Example: Find the coordinates of the maximum and minimum points for the curve .

Solution: At a turning point, 0. So = 0. Factorsise: 3x(x - 3) - 1(x - 3) = 0 (3x - 1)(x - 3) = 0 Find y coordinates from : When , When x = 3, So coordinates are: .

To decide whether they are a maximum or minimum calculate the gradient at either side of the point. : x = 0, x = 0.5, Therefore a minimum.

x = 3: x = 2, x = 4, Therefore a maximum.

Sketch of graph:

Note: You can also use the 2nd derivative to decide whether a turning point is a maximum or a minimum: If

Equation of a tangent tells you the gradient of a curve. The gradient m of a tangent line at the point can be found from . The equation of the tangent is then .

Example 2: Find the equation of the tangent to the graph at x = 1.

Solution: ? this is used to find gradients When x = 1, .

When x = 1,

So equation of tangent is So, y = 2x - 1. Perpendicular lines Suppose 2 lines have gradients . These lines are perpendicular if , i.e. . Equation of a normal To find the equation of a normal at the point : . Find the gradient from ; . Find the gradient m of the normal using ; . The equation of the normal is