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प्रश्न
Find the second order derivative of the function.
e6x cos 3x
उत्तर
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)`
`= - 3 e^(6x) sin 3 x + 6 e^(6x) cos 3 x`
`= e^(6x) (6 cos 3 x - 3 sin 3 x)`
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] xx e^(6x) * 6`
`= e^(6x) [- 18 sin 3x - 9 cos 3 x] + e^(6x) [36 cos 3x - 18 sin 3x]`
`= e^(6x) [- 36 sin 3x + 27 cos 3x]`
`= 9 e^(6x) [3 cos 3x - 4 sin 3x]`
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