Question

Find the second order derivative of the function.

e^6x cos 3x

Answer 1

Let, y = e^6x cos 3x

Differentiating both sides with respect to x,

`dy/dx = e^(6x) d/dx cos 3 x + cos 3 x d/dx e^(6x)`

= `e^(6x) (- sin 3 x) d/dx (3x) + cos 3 x * e^(6x) d/dx (6x)`

= −3e^6x sin 3x + 6e^6x cos 3x

= e^6x (6 cos 3x − 3 sin 3x)

Differentiating both sides again with respect to x,

`(d^2 y)/dx^2 = e^(6x) d/dx (6 cos 3 x - 3 sin 3 x) + (6 cos 3 x - 3 sin 3 x) d/dx e^(6x)`

= `e^(6x) [6 (- sin 3x) d/dx (3x) - 3 cos 3x d/dx (3x)] + [6 cos 3x - 3 sin 3x]e^(6x) d/dx (6x)`

= e^6x [−6 sin 3x · 3 − 3 cos 3x · 3] + [6 cos 3x − 3 sin 3x] × e^6x ⋅ 6

= e^6x [−18 sin 3x − 9 cos 3x] + e^6x [36 cos 3x − 18 sin 3x]

= e^6x [−36 sin 3x + 27 cos 3x]

= 9e^6x [3 cos 3x − 4 sin 3x]