Differentiation

Gradient of a tangent

A gradient on a curve can be calculated at any point, this is called differentiation.

As a chord gets shorted it gets closer to the tangent.

The gradient of y = x^2 at a general point (x,y) is 2x.

The gradient of y = x^3 at a general point (x,y) is 3x^2.

Differentiating polynomials

In general, for a function y = x^n the formula for the gradient is dy/dx = nx^n-1

Another way of writing dy/dx for a function f(x) is f '(x)

Tangents and normals

The tangent to a curve at a given point is the straight line which just touches the curve at that point.

The normal to a curve is the straight line perpendicular to the tangent at the point.

If the gradient of the tangent = m then the gradient of the normal = -1/m

To find the equation of a tangent we need the gradient and a point on the line.

Then we can use y-y1 = m(x-x1) where m is the gradient and (x1,x2) is the point you are given.

Stationary points

The "tops" and "bottoms" are called turning points.

When dy/dx = 0 y has a stationary value.

At both maximum and minimum points dy/dx = 0

When a gradient goes from negative to zero to negative it is classed as a point of inflexion.

When a gradient goes from positive to zero to positive it is classed as a point of inflexion.

Application: using the second derivative

There is another way of identifying the type of stationary point.

Remember dy/dx was the rate that y was changing as x changed.

If we differentiate again we find the rate that the gradient is changing.

The second differential is written as d^2y/dx^2

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