The Product Rule In words: “Keep the first, differentiate the second” + “Keep the second, differentiate the first”

Examples:1.Differentiate

Examples:2.Differentiate Now watch this.

Examples:3.Differentiate Try this using “words”

The Quotient Rule In words: “Keep the denominator, differentiate the numerator” “Keep the numerator, differentiate the denominator” – Denominator 2

Examples:1.Differentiate

Examples:2.Differentiate Try this using “words”

Add a denominator here

Derivatives of New Functions Definitions: Reminder: continue

    

    

    

Derivative of Use the Quotient Rule now Proof:

Derivatives of Prove these and keep with your notes. Use chain rule or quotient rule

Example: Given thatshow that

Exponential and Logarithmic Functions Reminder: andare inverse to each other. They are perhaps the most important functions in the applications of calculus in the real world. Alternative notation: Two very useful results: Also: Practise changing from exp to log and vice-versa. Learn these!

    x y

Derivatives of the Exponential and Logarithmic Functions Proof of (ii) (i)(ii)

Examples:1.Differentiate Use the Chain Rule 2.Differentiate Use the Product Rule

3.Differentiate Use the Chain Rule 4.Differentiate Use the Quotient Rule

Note:In general Useful for reverse i.e. INTEGRATION

Higher Derivatives Given that f is differentiable, if is also differentiable then its derivative is denoted by. The two notations are: function 1 st derivative 2 nd derivative …… n th derivative f ……

Example: If, write down is first second and third derivatives and hence make a conjecture about its nth derivative. Conjecture:The n th derivative is

Rectilinear Motion If displacement from the origin is a function of time I.e. then v - velocity a - acceleration

Example:A body is moving in a straight line, so that after t seconds its displacement x metres from a fixed point O, is given by (a)Find the initial dislacement, velocity and acceleration of the body. (b)Find the time at which the body is instantaneously at rest.

Extreme Values of a Function Understand the following terms: Critical Points Local Extreme Values  Local maximum  Local minimum End Point Extreme Values  End Point maximum  End Point minimum See, MIA Mathematics 1, Pages 54 – 55

A B Consider maximum turning point A. Notice, gradient of for x in the neighbourhood of A is negative. The Nature of Stationary Points i.e. is negative Similarly, gradient of for x in the neighbourhood of B is positive. i.e. is positive Consider a curve and the corresponding gradient function

The Nature of Stationary Points Rule for Stationary Points and minimum turning point and maximum turning point and possibly a point of inflexion but must check using a table of signs

Example: Consider At S.P. Now what does look like? Notice no Point of Inflexion.

Global Extreme Values Understand the following terms: Global Extreme Values  Global maximum  Global minimum See, MIA Mathematics 1, Pages 58 – 59

Example: Find the coordinates and nature of the stationary point on the curve At S.P. is a Minimum Turning Point What does this curve look like?

x y

Optimisation Problems A sector of a circle with radius r cm has an area of 16 cm 2. Show that the perimeter P cm of the sector is given by (a)(a) (b)(b)Find the minimum value of P. (a)(a) l r r now

(b)(b) At SP r = 4 gives a minimum stationary value of