Find the n th derivative of the following: 1 3 x - 5

Let y = 1 3 x - 5 Differentiating both sides w.r.t. x we get d y d x = - 1 . 3 3 x - 5 2 ⇒ d y d x = - 1 1 1 ! x 2 Again differentiating both sides w.r.t. x , we get d 2 y d x 2 = - 1 - 2 3 3 x - 5 - 3 . 3 ⇒ d 2 y d x 2 = - 1 2 . 3 2 . 2 ! 3 x - 5 3 Also, d 3 y d x 3 = - 1 2 . 3 2 2 ! - 3 3 x - 5 - 4 . 3 ⇒ d 3 y d x 3 = - 1 3 . 3 3 . 3 ! 3 x - 5 4 In general d n y d x n = - 1 n . 3 n . n ! 3 x - 5 n + !

Find the n th derivative of the following: y = e a x · cos b x + c

Let y = e a x · cos b x + c Differentiating both sides w.r.t. x we get d y d x = e a x - sin b x + c × b + a e a x cos b x + c ⇒ d y d x = e a x a cos b x + c - b sin b x + c ⇒ d y d x = e a x a 2 + b 2 a a 2 + b 2 cos b x + c - b a 2 + b 2 sin b x + c Let a a 2 + b 2 = cos α , then b a 2 + b 2 = sin α ⇒ d y d x = e a x a 2 + b 2 cos α cos b x + c - sin α sin b x + c ⇒ d y d x = e a x a 2 + b 2 1 2 cos α cos b x + c + α Again differentiating both sides w.r.t. x , we get d 2 y d x 2 = a 2 + b 2 1 2 e a x d d x cos b x + c + α + cos b x + c + α d d x e a x ⇒ d 2 y d x 2 = e a x a 2 + b 2 2 2 cos b x + c + 2 α Also, d 3 y d x 3 = e a x a 2 + b 2 3 2 cos b x + c + 3 α In general d n y d x n = e a x a 2 + b 2 n 2 cos b x + c + n α ⇒ d n y d x n = e a x a 2 + b 2 n 2 cos b x + c + n tan - 1 b a

Find the n th derivative of the following: y = e 8 x · cos 6 x + 7

Let y = e 8 x · cos 6 x + 7 Differentiating both sides w.r.t. x we get d y d x = e 8 x - sin 6 x + 7 × 6 + 8 e 8 x cos 6 x + 7 ⇒ d y d x = e 8 x 8 cos 6 x + 7 - 6 sin 6 x + 7 ⇒ d y d x = e 8 x 8 2 + 6 2 8 8 2 + 6 2 cos 6 x + 7 - 6 8 2 + 6 2 sin 6 x + 7 Let 8 8 2 + 6 2 = cos α , then 6 8 2 + 6 2 = sin α ⇒ d y d x = e 8 x 8 2 + 6 2 cos α cos 6 x + 7 - sin α sin 6 x + 7 ⇒ d y d x = e 8 x 8 2 + 6 2 1 2 cos α cos 6 x + 7 + α Again differentiating both sides w.r.t. x , we get d 2 y d x 2 = 8 2 + 6 2 1 2 e 8 x d d x cos 6 x + 7 + α + cos 6 x + 7 + α d d x e 8 x ⇒ d 2 y d x 2 = e 8 x 8 2 + 6 2 2 2 cos 6 x + 7 + 2 α Also, d 3 y d x 3 = e 8 x 8 2 + 6 2 3 2 cos 6 x + 7 + 3 α In general d n y d x n = e 8 x 8 2 + 6 2 n 2 cos 6 x + 7 + n α ⇒ d n y d x n = e 8 x 8 2 + 6 2 n 2 cos 6 x + 7 + n tan - 1 6 8