Methods for teaching differentiation and integration Understanding the chain rule, product rule and quotient rule Building a thorough understanding of differentiation techniques Answering tough calculus questions, gain an understanding of integration techniques Dealing with the multi-step, more wordy questions

Chain rule for composite functions To differentiate a function of a function, we use a new letter u to help us to write the function in two steps. We find the separate derivatives, and use the chain rule to help us combine them into the answer.

1. Differentiate Write Differentiate each – take care with the name of each derivative Combine derivatives together Encourage good students to do this in their head

2. Differentiate Write

Product and quotient rule Introduction – establish the need for a rule by demonstrating you can’t just multiply derivatives together Use your judgement about proving the rule from first principles!

Product rule y is given as the product of two functions of x. The derivative can be made out of the two separate derivatives using the formula This isn’t in the formula books!!

4.Differentiate Set out equations on the right of the page Differentiate each – take care with the names of the derivatives Combine together Simplify (the hard part!)

Quotient rule y is given as a fraction The derivative can be made out of the two separate derivatives using the formula This is in the formula books!! You can use it to help you remember the product rule as well – take the top with - replaced with +

8.Differentiate Set out equations on the right of the page Differentiate each – take care with the names of the derivatives Combine together Simplify

Which of the functions on the table are of this type? Integrals of the form Once you recognise the fraction as being of this type, you can just write down the answer. Which of the functions on the table are of this type?

Integration by substitution We use u to simplify what we have to integrate, but we must change the dx in terms of du as well as the limits. We differentiate the u = ..... equation and split the derivative

Example

Example

Integration by parts – as near as we get to a product rule for integration  We use the product rule for differentiation to get our formula We choose one part to be u and the other to be We can use the word LATE to help us choose – u is the one that comes first (Log, Algebra, Trig, Exponential)

Example

How to choose a method? It comes with practice! Try it and see! Look out for bits like which are too complicated to integrate Other ideas? Sorting cards for integrating methods – look for ways of asking extension questions in all of these paired or group activities

Forming and solving differential equations Newton’s law of cooling states that the rate of cooling of a body is proportional to the excess temperature. (a) Using write this as a differential equation. (b) My coffee is 95° when it is made and is initially cooling at 5° per minute when the room is 20°. Find an expression for θ at time t. (c) If I can drink the coffee at 65°, how long do I have to wait before I can drink it?

Forming and solving differential equations Newton’s law of cooling states that the rate of cooling of a body is proportional to the excess temperature. (a) Using write this as a differential equation. The minus indicates decrease in temperature The dt derivative is always the rate of …. Excess temperature The k converts “proportional to” into an equation

Forming and solving differential equations (b) My coffee is 95° when it is made and is initially cooling at 5° per minute when the room is 20°. Find an expression for θ at time t. You can evaluate c first. You can collect ln terms together before “e-ing”!!

Forming and solving differential equations (c) If I can drink the coffee at 65°, how long do I have to wait before I can drink it?