# A-level Mathematics/CIE/Pure Mathematics 1/Differentiation

## The Gradient of a Point on a Curve[edit]

The **gradient** of a point on a curve is equal to the gradient of the *tangent line* at that point. Since it is difficult to measure the gradient of the tangent directly, we use *secant lines* which approach the tangent. A secant line is a line that travels between two points on the curve.

### The Formal Definition of a Derivative[edit]

Suppose we want to find the gradient of a point at coordinates . We can approximate it using a secant line that travels through and where is a small change in and is the small change in that results from it.

The gradient of the secant line is . As gets smaller and smaller, approaches the gradient of the point .

The *limit* of as approaches zero is the value that we are looking for: the gradient of the point . For an arbitrary point , the gradient can be given by the derivative.

The *derivative* of a function is a function which provides the gradient of a point on the curve produced by . The derivative is formally defined as

### Notation[edit]

There are two main ways to write the derivative of a function: and . These both mean the same thing: the derivative of with respect to .

## Simple Rules[edit]

These are a few simple rules that make differentiation of complicated expressions easier.

### The Power Rule[edit]

e.g.

### The Sum Rule[edit]

e.g.

### The Constant Rule[edit]

e.g.

### The Chain Rule[edit]

e.g.

## Tangents & Normals[edit]

A **tangent** is a line which travels through a given point and has a gradient which is equal to the gradient of the curve at that point. A **normal** is a line which travels through a given point and is perpendicular to the tangent of the curve at that point.

To find the equation of the tangent or normal of a curve at a particular point, we can use the equations of a line that we used in Coordinate Geometry.

e.g. Find the equation of the tangent and normal of a curve at the point

## Increasing & Decreasing Functions[edit]

An **increasing function** is a function whose gradient is always greater than or equal to zero.

A **decreasing function** is a function whose gradient is always less than or equal to zero.

## Rates of Change[edit]

The **rate of change** of a quantity is the derivative of a quantity with respect to time. For instance, if water were flowing into a bucket at a rate of 1 litre per second, the rate of change of the volume of water in the bucket would be 1 litre per second.

In some cases, the rates of change of two quantities are connected. For example, a circle with a changing radius will have a changing area that depends on the radius. The area of a circle is related to the radius by . If the radius is increasing at a rate of 3 cm/s, its rate of change . The rate of change of area can be found using the chain rule:

If the radius is increasing at a rate of 3 cm/s, and is currently 5cm, the rate of change of area is cm^{2}/s.

## Stationary Points[edit]

A **stationary point** is a point on the curve at which the gradient is zero. This means that the derivative of the function at that point is equal to zero.

Stationary points are either *maxima* or *minima*. Maxima are where the function reaches a maximum value. Minima are where the function reaches a minimum value. We can determine whether a stationary point is a maximum or minimum by looking at the second derivative.

If the second derivative is positive, the gradient is increasing with input, and thus the stationary point is a minimum.

If the second derivative is negative, the gradient is decreasing with input, and thus the stationary point is a maximum.

### Curve Sketching[edit]

Stationary points are useful when sketching graphs of curves. By marking the stationary points, we can draw graphs more accurately than otherwise.