Differentiation

The full A-level topic: gradients, the power rule, tangents, stationary points and first principles. Built for KS5.

Notation:

The gradient of a chord

Differentiation is all about gradient. A chord is a straight line joining two points on a curve. The closer the points get, the closer the chord gradient gets to the true gradient of the curve at that point.

Zoom in on the chord: 1x
curve y = f(x) chord drag to move

As you zoom in, notice the chord and the curve almost lie on top of one another - but they are never exactly the same. The chord is straight; the curve bends. Differentiation finds what the chord gradient tends towards as the gap shrinks to nothing.

The idea behind the rule - differentiation from first principles (informal)

Instead of two specific numbers, use two general points. Let the first be at and the second a small distance further along, at . Their -values are then and .

  1. Gradient of the chord is rise over run:
  2. Shrink the gap. Make smaller and smaller. The chord rotates towards the tangent at A - but is never actually zero.
  3. Take the limit. The value the chord gradient tends towards as is called the derivative:

This is the idea behind differentiation from first principles. The full proof with worked examples is on the First Principles tab.

The rule for differentiating

For any power of there are just two steps.

To differentiate a power of x
  • Multiply by the power.
  • Reduce the power by one.
The power 1: y = x

Multiply by 1, drop the power to 0. Since , the simply disappears - the gradient of is always 1.

The power 0: a constant

A constant is . Multiplying by the power 0 wipes it out - every constant differentiates to 0 (a flat line has zero gradient).

Sums: differentiate term by term

Apply the rule to each term separately - the sum and difference rule means you never have to deal with a whole expression at once.

Differentiation notation - , and

Lagrange notation - . The derivative of is written ("f-dash x"). The second derivative is - covered on the Stationary Points tab.

Leibniz notation - . If the function is written , its derivative is ("dee-y by dee-x") - the rate of change of with respect to . It comes directly from the chord gradient letting . The second derivative is .

Operator notation - . On its own, is an instruction: "differentiate whatever follows with respect to ." So .

Worked examples

The key step is almost always to rewrite as a power of before applying the rule.

Questions

Sketching a gradient function

You can sketch the shape of from the graph of without any algebra. Reveal the steps in order.

Stationary point

A point where the gradient is zero, - the curve is momentarily flat. On the gradient function these appear as zeros (the graph crosses or touches the -axis).

Reveal the steps in order
y = f(x)
y = f′(x)
  1. Mark the stationary points. Find every place the curve is flat (). Directly below each one, the gradient function sits on the -axis.
  2. Positive or negative gradient? Uphill (left to right) means - the gradient graph is above the axis. Downhill means - it is below.
  3. Is the gradient increasing or decreasing? Where the curve steepens, is rising; where it flattens, is falling. This fixes the shape between the axis crossings.

Worked examples

Each box shows the original curve and its gradient function side by side with the step-by-step reasoning.

Increasing & decreasing functions

A function is increasing where and decreasing where . Select a function, toggle the highlights, then read the working.

Choose a function
increasing decreasing turning point

Tangents & normals

curve tangent normal

Drag the point along .

What is a tangent? What is a normal?

Tangent: the straight line that just touches the curve at a point and has the same gradient as the curve there. Its gradient is .

Normal: the straight line at the same point that is perpendicular to the tangent.

Stationary points

A stationary point has . To classify it we look at the second derivative - the gradient of the gradient.

Types of stationary point

When the test is inconclusive - the point may or may not be a point of inflection. See the Points of Inflection tab.

Practice

Points of inflection

A point of inflection is where the curve changes the way it bends - from concave to convex or vice versa. The second derivative is zero here, but not all zeros of the second derivative are points of inflection.

When the max/min test is inconclusive. The point may be a point of inflection - or it may still be a minimum or maximum. You must look further.
The reliable fallback - the gradient table
Not every point of inflection is stationary. An inflection is where and changes sign. That can happen mid-slope where . See the picture.

Practice

The gradient function

The derivative is itself a function - it gives the gradient of the curve at every value of . Drag the point along the left graph and watch the right graph track the gradient at that point.

y = f(x)
y = f′(x) (gradient function)
What the two graphs are telling you

Differentiation from first principles

Every differentiation rule comes from this definition. We take two points on , find the gradient of the chord, and ask what value that gradient tends towards as the gap closes.

Choose a function
2 Make h smaller - watch the chord tend towards the tangent
curve chord gradient derivative (tangent)

Drag the blue point to change .

The limit notation

means "the value this expression gets arbitrarily close to as approaches 0." We never actually set (we cannot divide by zero) - we ask what the expression tends towards. In that limit the chord gradient tends to .

Worked Solution