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Question
Find the second order derivative of the function.
e6x cos 3x
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Solution
Let, y = e6x 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)`
= −3e6x sin 3x + 6e6x cos 3x
= e6x (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)`
= e6x [−6 sin 3x · 3 − 3 cos 3x · 3] + [6 cos 3x − 3 sin 3x] × e6x ⋅ 6
= e6x [−18 sin 3x − 9 cos 3x] + e6x [36 cos 3x − 18 sin 3x]
= e6x [−36 sin 3x + 27 cos 3x]
= 9e6x [3 cos 3x − 4 sin 3x]
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