𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 Derivative sin x, cos x 6A

Trig Differentiation Starter: Sketch the graphs of sinx, cos x tan x KUS objectives BAT differentiate trig functions using what we have learned so far BAT differentiate mixed trig, exponential, logarithm and other functions Starter: Sketch the graphs of sinx, cos x tan x Where are the stationary points on sinx, cos x tan x ? Sketch graphs of arcsin x, arccos x, arctan x

The gradient function of: Gradients I: Trig Graphs The gradient function of: y = sinx y = cosx dy/dx= cosx dy/dx= -sinx

WB 1 Differentiate each of : a) 𝑦= sin 3𝑥 b) 𝑦= sin 2 3 𝑥 c) 𝑦= 𝑠𝑖𝑛 2 𝑥 d) 𝑦= 𝑐𝑜𝑠 2 𝑥 e) y=cos 3𝑥 2 −8 𝑎) 𝑦=𝑠𝑖𝑛3𝑥 d) 𝑦=( cos 𝑥 ) 2 𝑑𝑦 𝑑𝑥 =3 cos 3𝑥 𝑑𝑦 𝑑𝑥 =2 ( cos 𝑥 ) 1 − sin 𝑥 =−2 sin 𝑥 cos 𝑥 =− sin 2𝑥 b) 𝑦=𝑠𝑖𝑛 2 3 𝑥 e) 𝑦=cos 3𝑥 2 −8 𝑑𝑦 𝑑𝑥 = 2 3 𝑐𝑜𝑠 2 3 𝑥 𝑑𝑦 𝑑𝑥 =−sin 3𝑥 2 −8 ×(6𝑥) 𝑑𝑦 𝑑𝑥 =−6𝑥 sin 3𝑥 2 −8 𝑐) 𝑦=(𝑠𝑖𝑛𝑥 ) 2 𝑑𝑦 𝑑𝑥 =2(𝑠𝑖𝑛𝑥 ) 1 (𝑐𝑜𝑠𝑥) 𝑑𝑦 𝑑𝑥 =2𝑠𝑖𝑛𝑥𝑐𝑜𝑠𝑥=sin 2x

WB 2 Differentiate each of : a) 𝑦= 𝑐𝑜𝑠 (4𝑥−3) b) 𝑦= 𝑐𝑜𝑠 3 𝑥 c) 𝑦= 1 2 sin 2𝑥 2 +2𝑥 d) 𝑦= 𝑠𝑖𝑛 2 𝑥 2 e) y= 1 2 𝑐𝑜𝑠 2𝑥+1 2 𝑑) 𝑦= sin 𝑥 2 2 𝑎) 𝑦=𝑐𝑜𝑠⁡(4𝑥−3) 𝑑𝑦 𝑑𝑥 =−4𝑠𝑖𝑛⁡(4𝑥−3) 𝑑𝑦 𝑑𝑥 =2 sin 𝑥 2 1 1 2 cos 𝑥 2 𝑑𝑦 𝑑𝑥 = sin 𝑥 2 cos 𝑥 2 = 1 2 sin 𝑥 2 𝑏) 𝑦=(𝑐𝑜𝑠𝑥 ) 3 𝑑𝑦 𝑑𝑥 =3(𝑐𝑜𝑠𝑥 ) 2 (− sin 𝑥 ) e) 𝑦= 1 2 cos 2𝑥+1 2 𝑑𝑦 𝑑𝑥 =−3𝑐𝑜 𝑠 2 𝑥 𝑠𝑖𝑛𝑥 𝑑𝑦 𝑑𝑥 =− 1 2 sin 2𝑥+1 2 ×2(2𝑥+1) c) 𝑦= 1 2 sin 2𝑥 2 +2𝑥 𝑑𝑦 𝑑𝑥 = 2x+1 sin 2𝑥+1 2 𝑑𝑦 𝑑𝑥 = 1 2 cos 2𝑥 2 +2𝑥 ×(4𝑥+2) 𝑑𝑦 𝑑𝑥 = 2x+1 sin 2𝑥 2 +2𝑥

Product and quotient rules Derivative tan x Product and quotient rules If: 𝑦=𝑡𝑎𝑛𝑥 𝑑𝑦 𝑑𝑥 =𝑠𝑒 𝑐 2 𝑥 Then: If: 𝑦=𝑡𝑎𝑛⁡𝑓 𝑥 𝑑𝑦 𝑑𝑥 = 𝑓 ′ 𝑥 𝑠𝑒 𝑐 2 𝑓(𝑥) Then:

WB 3ab Differentiate each of : a) 𝑦= 𝑥 2 cos 𝑥 b) 𝑦=4𝑥 sin 2𝑥 c) 𝑦= 1 2 sin 𝑥 cos 2𝑥 d) 𝑦= sin 3𝑥 cos 𝑥 a) 𝑦= 𝑥 2 cos 𝑥 Use the product rule WORKING OUT SPACE 𝑑𝑦 𝑑𝑥 = cos 𝑥 × 2𝑥 + 𝑥 2 ×− sin 𝑥 𝑢= 𝑥 2 𝑣= cos 𝑥 𝑑𝑦 𝑑𝑥 =2𝑥 cos 𝑥 − 𝑥 2 sin 𝑥 𝑑𝑢 𝑑𝑥 =2𝑥 𝑑𝑣 𝑑𝑥 =− sin 𝑥 b) 𝑦= 4𝑥 sin 2𝑥 Use the product rule WORKING OUT SPACE 𝑑𝑦 𝑑𝑥 = sin 2𝑥 × 4 + 4𝑥 ×2 cos 2𝑥 𝑢=4𝑥 𝑣= sin 2𝑥 𝑑𝑦 𝑑𝑥 =4 sin 2𝑥 +8𝑥 cos 2𝑥 𝑑𝑢 𝑑𝑥 =4 𝑑𝑣 𝑑𝑥 =2 cos 2𝑥

WB 3cd Differentiate each of : a) 𝑦= 𝑥 2 cos 𝑥 b) 𝑦=4𝑥 sin 2𝑥 c) 𝑦= 1 2 sin 𝑥 cos 2𝑥 d) 𝑦= sin 3𝑥 cos 𝑥 c) 𝑦= 1 2 sin 𝑥 cos 2𝑥 Use the product rule WORKING OUT SPACE 𝑢= sin 𝑥 𝑣= 1 2 cos 2𝑥 𝑑𝑦 𝑑𝑥 = 1 2 cos 2𝑥 × cos 𝑥 + sin 𝑥 × − sin 2𝑥 𝑑𝑦 𝑑𝑥 = 1 2 cos x cos 2𝑥 − sin 𝑥 sin 2𝑥 𝑑𝑢 𝑑𝑥 = cos 𝑥 𝑑𝑣 𝑑𝑥 =− sin 2𝑥 c) 𝑦= sin 3𝑥 cos 𝑥 Use the product rule WORKING OUT SPACE 𝑑𝑦 𝑑𝑥 = cos 𝑥 × 3 cos 3𝑥 + sin 3𝑥 × − sin 𝑥 𝑢= sin 3𝑥 𝑣= cos 𝑥 𝑑𝑦 𝑑𝑥 =3cos x cos 2𝑥 − sin 𝑥 sin 3𝑥 𝑑𝑢 𝑑𝑥 =3 cos 3𝑥 𝑑𝑣 𝑑𝑥 =− sin 𝑥

WB 4ab Differentiate each of : a) 𝑦= sin 10𝑥 5𝑥 b) 𝑦= 𝑥 2 cos 𝑥 WORKING OUT SPACE Use the quotient rule 𝑢= sin 10𝑥 𝑣=5𝑥 𝑑𝑦 𝑑𝑥 = 5𝑥× 10 cos 10𝑥 − sin 10𝑥 ×5 5𝑥 2 𝑑𝑢 𝑑𝑥 =10 cos 10𝑥 𝑑𝑣 𝑑𝑥 =5 𝑑𝑦 𝑑𝑥 = 10𝑥 cos 10𝑥 − sin 10𝑥 5 𝑥 2 b) 𝑦= 𝑥 2 cos 𝑥 WORKING OUT SPACE Use the quotient rule 𝑢= 𝑥 2 𝑣= cos 𝑥 𝑑𝑦 𝑑𝑥 = cos 𝑥 × 2𝑥 − 𝑥 2 × − sin 𝑥 cos 𝑥 2 𝑑𝑢 𝑑𝑥 =2𝑥 𝑑𝑣 𝑑𝑥 =− sin 𝑥 𝑑𝑦 𝑑𝑥 = 2𝑥 cos 𝑥 + 𝑥 2 sin 𝑥 𝑐𝑜𝑠 2 𝑥

Quotients Rule! 𝑑𝑦 𝑑𝑥 = cos 𝑥 × cos 𝑥 − sin 𝑥 × − sin 𝑥 cos 𝑥 2 WB 5 Find the derivative of tan x 𝑦= sin 𝑥 cos 𝑥 𝑑𝑦 𝑑𝑥 = 𝑣 𝑑𝑢 𝑑𝑥 −𝑢 𝑑𝑣 𝑑𝑥 𝑣 2 Use the quotient rule WORKING OUT SPACE 𝑢= sin 𝑥 𝑣= cos 𝑥 𝑑𝑦 𝑑𝑥 = cos 𝑥 × cos 𝑥 − sin 𝑥 × − sin 𝑥 cos 𝑥 2 𝑑𝑢 𝑑𝑥 = cos 𝑥 𝑑𝑣 𝑑𝑥 =− sin 𝑥 𝑑𝑦 𝑑𝑥 = 𝑠𝑖𝑛 2 𝑥 + − 𝑐𝑜𝑠 2 𝑥 𝑐𝑜𝑠 2 𝑥 𝑑𝑦 𝑑𝑥 = 1 𝑐𝑜𝑠 2 𝑥 = 𝑠𝑒𝑐 2 𝑥 𝑑 𝑑𝑥 tan 𝑥 = 𝑠𝑒𝑐 2 𝑥

WB 6 Differentiate each of : a) 𝑦=𝑡𝑎 𝑛 4 𝑥 b) 𝑦=𝑥 tan 2𝑥 𝑑𝑦 𝑑𝑥 =4(𝑡𝑎𝑛𝑥 ) 3 (𝑠𝑒 𝑐 2 𝑥) 𝑑𝑦 𝑑𝑥 =4𝑡𝑎 𝑛 3 𝑥𝑠𝑒 𝑐 2 𝑥 b) 𝑦=𝑥 𝑡𝑎𝑛2𝑥 WORKING OUT SPACE Use the product rule 𝑑𝑦 𝑑𝑥 =𝑥×(2𝑠𝑒 𝑐 2 2𝑥) +(𝑡𝑎𝑛2𝑥)×1 𝑢=𝑥 𝑣=𝑡𝑎𝑛2𝑥 𝑑𝑢 𝑑𝑥 =1 𝑑𝑣 𝑑𝑥 =2𝑠𝑒 𝑐 2 2𝑥 𝑑𝑦 𝑑𝑥 =2𝑥𝑠𝑒 𝑐 2 2𝑥+ 𝑡𝑎𝑛2𝑥

𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝑒𝑐𝜃= 1 𝑠𝑖𝑛𝜃 Derivative Mixed functions

𝑑𝑦 𝑑𝑥 = 𝑠𝑖𝑛𝑥× 1 𝑥 − ln 𝑥 × cos 𝑥 sin 𝑥 2 WB 7ab Differentiate each of : a) 𝑦= ln 𝑥 sin 𝑥 b) 𝑦= 𝑒 𝑥 sin 𝑥 a) 𝑦= ln 𝑥 sin 𝑥 𝑑𝑦 𝑑𝑥 = 𝑣 𝑑𝑢 𝑑𝑥 −𝑢 𝑑𝑣 𝑑𝑥 𝑣 2 Use the quotient rule WORKING OUT SPACE 𝑑𝑦 𝑑𝑥 = 𝑠𝑖𝑛𝑥× 1 𝑥 − ln 𝑥 × cos 𝑥 sin 𝑥 2 𝑢=𝑙𝑛𝑥 𝑣=𝑠𝑖𝑛𝑥 𝑑𝑢 𝑑𝑥 = 1 𝑥 𝑑𝑣 𝑑𝑥 =𝑐𝑜𝑠𝑥 𝑑𝑦 𝑑𝑥 = 𝑠𝑖𝑛𝑥 − 𝑥 ln 𝑥 cos 𝑥 𝑥 𝑠𝑖𝑛 2 𝑥 b) 𝑦= 𝑒 𝑥 sin 𝑥 Use the product rule WORKING OUT SPACE 𝑑𝑦 𝑑𝑥 =𝑠𝑖𝑛𝑥× 𝑒 𝑥 + 𝑒 𝑥 × cos 𝑥 𝑢= 𝑒 𝑥 𝑣=𝑠𝑖𝑛𝑥 𝑑𝑢 𝑑𝑥 = 𝑒 𝑥 𝑑𝑣 𝑑𝑥 =𝑐𝑜𝑠𝑥 𝑑𝑦 𝑑𝑥 = 𝑒 𝑥 sin 𝑥 + cos 𝑥

𝑑𝑦 𝑑𝑥 = 𝑙𝑛× 𝑠𝑒𝑐 2 𝑥 − tan 𝑥 × 1 𝑥 tan 𝑥 2 WB 7cd Differentiate each of : c) 𝑦= 𝑒 𝑥 2 sin 𝑥 2 d) 𝑦= tan 𝑥 ln 𝑥 c) 𝑦= 𝑒 𝑥 sin 𝑥 Use the product rule WORKING OUT SPACE 𝑑𝑦 𝑑𝑥 = sin 𝑥 2 × 2𝑥 𝑒 𝑥 2 + 𝑒 𝑥 2 × 1 2 cos 𝑥 2 𝑢= 𝑒 𝑥 2 𝑣= sin 𝑥 2 𝑑𝑢 𝑑𝑥 =2𝑥 𝑒 𝑥 2 𝑑𝑣 𝑑𝑥 = 1 2 cos 𝑥 2 𝑑𝑦 𝑑𝑥 = 𝑒 𝑥 2 2𝑥 sin 𝑥 2 + 1 2 cos 𝑥 d) 𝑦= tan 𝑥 ln 𝑥 Use the quotient rule WORKING OUT SPACE 𝑑𝑦 𝑑𝑥 = 𝑙𝑛× 𝑠𝑒𝑐 2 𝑥 − tan 𝑥 × 1 𝑥 tan 𝑥 2 𝑢= tan 𝑥 𝑣= ln 𝑥 𝑑𝑢 𝑑𝑥 = 𝑠𝑒𝑐 2 𝑥 𝑑𝑣 𝑑𝑥 = 1 𝑥 𝑑𝑦 𝑑𝑥 = 𝑥 ln 𝑥 𝑠𝑒𝑐 2 𝑥− tan 𝑥 𝑥 𝑥 𝑡𝑎𝑛 2 𝑥

self-assess using: R / A / G ‘I am now able to ____ . KUS objectives BAT differentiate trig functions using what we have learned so far BAT differentiate mixed trig, exponential, logarithm and other functions self-assess using: R / A / G ‘I am now able to ____ . To improve I need to be able to ____’