Dr Duncan James > UPmaths > Calculus

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Differentiation and Integration (Calculus)

When I teach this topic I focus three main things:- (i) differentiation and integration are opposites; (ii) differentiation finds the gradient and; (iii) integration finds area.

UP1, UP2, UP3 and UP4 are described on the levels page.

"Where do mathematicians go sailing? In the Sea of Integration."


Differentiation and integration are not part of UP1 nor UP2.

Differentiation Introduction

At level UP3 the differentiation of polynomials is typically introduced. Polynomials are things like x2, x5 and 11x3. This tells us about how something is changing.

Differentiation is often first taught as a process without understanding how it is used.

Integration Introduction

At level UP3 the integration of polynomials is often introduced at the same time as differentiation. It is literally the opposite of differentiation.

Rate of Change

Applied differentiation questions often involve rate of change.

The Differential is the Gradient of a Curve

Applied differentiation questions often use it as a tool to find the gradient of a curve.

The idea that differentiation is the gradient of a graph/curve is an important stage in learning as this is an important application at higher levels. This means the original equation of the curve can be deduced from the gradient. It also means lots of applied problems involving things like tangents can be solved.

The Second Differential

If you apply differentiation you get the rate-of-change of the rate-of-change! This can be thought of as the acceleration.

Integration Represents Area

Integration can be used to find the area between a curve and an axis of a graph. This is another popular type of question.

Mixed Calculus Questions

If you are getting comfortable with differentiation and integration and how it is used then you might like to try a range of questions using all these ideas.

Proof of How Differentiation Works

You might be interested to see how we can "prove" that differentiation gives the gradient of a curve. Sometimes a course requires you to understand this.

Differentiation of y if a Function or Constant (YouTube)

Differentiation of Other Functions

Many other functions can be differentiated.

Integration of Other Functions

And obviously many other functions can also be integrated.

Integration Overview for UP4 (YouTube)

Parametric Calculus

A parametric function can represent graphs where one value of x has multiple values of y; there are other benefits as well. The ability to differentiate and integrate these functions is useful.

Here is a link back to the UPmaths homepage.