Differentiation and IntegrationDifferentiation and Integration together are called Calculus.

DifferentiationDifferentiate if the question says:Derive Find the derivativeFind the rate of changeFind or f (x)Maximise or minimiseHow do you know when to differentiate?

Differentiation

Change to Get rid of fractions with x on the bottom:eg

Multiply out bracketsHow do you prepare for differentiation?

DifferentiationHow do you differentiate?Bring the power to the front (to multiply)Subtract 1 from the power

Differentiation in real lifeWhy was differentiation developed originally?In Isaac Newton's day, shipwrecks occurred regularly because the ship was not where the captain thought it was using only the stars for guidance. Calculus was developed to improve navigation techniques.

Differentiation in real lifeDifferentiation is used in economics and business to find optimum solutions.

eg to use the minimum amount of material to make the packaging, or to maximise efficiency of products or to minimise their costs.

Differentiation and Integration

Click for calculus video

Graphs of the derivativeIf you are asked to draw the graph of the derivative, keep in mind that this is a graph of the gradient.Start by marking points on the axes where there are turning points on the original graph.Now look at the gradient before and after each of these points: Is it up (in which case, draw above the x- axis) or down (draw below x-axis)?Make sure you curve is smooth

IntegrationIntegrate if the question says:Find the integralFind the area under a curveFind the area between 2 curvesFind Find y when you are given How do you know when to integrate?

Integration

Change to Get rid of fractions with x on the bottom:eg

Multiply out bracketsHow do you prepare for integration?

IntegrationHow do you integrate?Add 1 to the powerDivide by the new power+ c

Integration in real lifeIntegration is used to analyse crash test data, to improve safety of cars.

It is also used by engineers to find the force of water from the surface area of a reservoir, to design dams.

Further Calculus

sin ax a cos ax cos ax -a sin ax

sin ax - cos ax cos ax sin ax

differentiates tointegrates toTo differentiate or integrate a trig function,use the formula sheet:

Further CalculusTo differentiate or integrate a bracket with a power:1.Think of the whole bracket as an x anddifferentiate or integrate it as usual

2.Multiply by the derivative of the bracket(if differentiating)or Divide by the derivative of the bracket(if integrating)

Integration

Click for an integration joke