Aim: Chain rules with trigonometric function

Do Now:

1.write 𝑓 (𝑔 (𝑥 ))

2. find 𝑓 ′(𝑥 )

3. find 𝑔 ′ (𝑥)

Let x

𝑦 ′= 𝑓 ′ [𝑔 (𝑥 ) ] ∙𝑔′ (𝑥 )=5 𝑠𝑖𝑛4 𝑥𝑐𝑜𝑠𝑥

𝑦=sin (𝑥2+𝑥 )

𝑑𝑑𝑥

sin (𝑥2+𝑥 )=¿

outside

inside

cos (𝑥2+𝑥)∙ (2𝑥+1)

Derivative of outside

Inside left alone

Derivative of inside

𝑦=𝑡𝑎𝑛√𝑥

𝑑𝑑𝑥

𝑡𝑎𝑛√𝑥

outside

inside

¿ 𝑠𝑒𝑐2 √𝑥 ∙ 12√𝑥

Derivative of outside

Inside left alone

Derivative of inside

¿ 𝑠𝑒𝑐2√𝑥

2√𝑥

𝑑𝑑𝑥

𝑐𝑜𝑠23 𝑥¿2 cos 3𝑥 ∙𝑑𝑑𝑥

¿¿

We sometimes have to use chain rule two or three times to get the job done. Here is the example

(3x)

¿2 cos 3𝑥 ¿¿¿−6 cos3 𝑥 sin 3𝑥

𝑑𝑑𝑥

sin (1+ tan 2𝑥 )¿cos (1+𝑡𝑎𝑛2𝑥 ) 𝑑𝑑𝑥

(1+𝑡𝑎𝑛2 𝑥)

¿2 cos (1+𝑡𝑎𝑛2𝑥 ) ∙𝑠𝑒𝑐22 𝑥

Find𝑑𝑦𝑑𝑥

1. 𝑦=sin (7−5 𝑥)

2. 𝑦=cos (−𝑥3

)

3. 𝑦=𝑠𝑖𝑛3 4 𝑥

4. 𝑦=cos (2−cot 3 𝑥)

−𝟓𝐬𝐢𝐧 (𝟕−𝟓 𝒙)

𝟏𝟑𝐬𝐢𝐧 (

− 𝒙𝟑

)

𝟏𝟐 𝒔𝒊𝒏𝟐𝟒 𝒙 𝒄𝒐𝒔𝟒 𝒙

𝟑𝐬𝐢𝐧 (𝟐−𝒄𝒐𝒕𝟑 𝒙)(𝒄𝒔𝒄𝟐𝟑 𝒙)