Miscellaneous

Misc 1 (ii) Important

Misc 1 (iii)

Misc 1 (iv) Important

Misc 2

Misc 3 Important

Misc 4 Important

Misc 5

Misc 6 Important

Misc 7

Misc 8 Important

Misc 9 Important

Misc 10

Misc 11

Misc 12 Important

Misc 13

Misc 14 Important

Misc 15

Misc 16

Misc 17 Important You are here

Misc 18 Important

Misc 19

Misc 20 Important

Misc 21

Misc 22 Important

Misc 23

Misc 24 Important

Misc 25

Misc 26

Misc 27 Important

Misc 28 Important

Misc 29 Important

Misc 30 Important

Last updated at May 29, 2023 by Teachoo

Misc 17 Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): sin〖x + cosx 〗/sin〖x − cosx 〗 Let f (x) = sin〖x + cosx 〗/sin〖x − cosx 〗 Let u = sin x + cos x & v = sin x – cos x ∴ f(x) = 𝑢/𝑣 So, f’(x) = (𝑢/𝑣)^′ Using quotient rule f’(x) = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 Finding u’ & v’ u = sin x + cos x u’ = (sin x + cos x)’ = (sin x)’ + (cos x)’ = cos x – sin x v = sin x – cos x v’= (sin x – cos x)’ = (sin x)’ – (cos x)’ = cos x – ( – sin x) = cos x + sin x Derivative of sin x = cos x Derivative of cos x = – sin x Now, f’(x) = (𝑢/𝑣)^′ = (𝑢^′ 𝑣 −〖 𝑣〗^′ 𝑢)/𝑣^2 = ( (cos〖𝑥 −〖 sin〗〖𝑥) (sin〖𝑥 −〖 cos〗〖𝑥) − (cos〖𝑥 +〖 sin〗〖𝑥) (sin〖𝑥 +〖 cos〗〖𝑥)〗 〗 〗 〗 〗 〗 〗 〗)/〖(sin〖x −co𝑠 𝑥〗)〗^2 = (−(sin〖𝑥 −〖 cos〗〖𝑥) (sin〖𝑥 −〖 cos〗〖𝑥) − (sin〖𝑥 + cos〖𝑥) (sin〖𝑥 +〖 cos〗〖𝑥)〗 〗 〗 〗 〗 〗 〗 〗)/〖(sin〖x − co𝑠 𝑥〗)〗^2 = (〖−(sin〖x − co𝑠 𝑥〗)〗^2 − 〖(sin〖x + co𝑠 𝑥〗)〗^2)/〖(sin〖x − co𝑠 𝑥〗)〗^2 Using (a + b)2 = a2 + b2 + 2ab (a – b)2 = a2 + b2 – 2ab = ( − [(sin2〖𝑥 +〖 cos2〗〖𝑥 − 2 sin〖𝑥 〖 cos〗〖𝑥) + (𝑠𝑖𝑛2𝑥 + 𝑐𝑜𝑠2𝑥 + 2𝑠𝑖𝑛𝑥 cos〖𝑥)]〗 〗 〗 〗 〗)/〖(sin〖x − co𝑠 𝑥〗)〗^2 = ( − ( 2𝑠𝑖𝑛2𝑥 + 2𝑐𝑜𝑠2𝑥 − 0))/〖(sin〖x − co𝑠 𝑥〗)〗^2 = ( −2 (𝒔𝒊𝒏𝟐𝒙 + 𝒄𝒐𝒔𝟐𝒙))/〖(sin〖x − co𝑠 𝑥〗)〗^2 = ( −2 (𝟏))/〖(sin〖x − co𝑠 𝑥〗)〗^2 = ( −𝟐 )/〖(𝒔𝒊𝒏〖𝐱 − 𝒄𝒐𝒔 𝒙〗)〗^𝟐 (Using sin 2 x + cos 2 x = 1)