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Question
Find the second order derivatives of the following function e6x cos 3x ?
Solution
We have,
\[y = e^{6x} \cos 3x\]
\[\text { Differentiating w . r . t . x, we get }\]
\[\frac{d y}{d x} = e^{6x} \times 6 \times \cos 3x + e^{6x} ( - \sin 3x \times 3)\]
\[ = 6 e^{6x} \cos3x - 3 e^{6x} \sin 3x\]
\[\text { Differentiating again w . r . t . x, we get }\]
\[\frac{d^2 y}{d x^2} = 6 e^{6x} \cos3x \times 6 - 6 e^{6x} \sin3x \times 3 - 3 \times 6 e^{6x} \sin3x - 3 e^{6x} \times 3 \cos 3x\]
\[ = 27 e^{6x} \cos3x - 36 e^{6x} \sin3x\]
\[ = 9 e^{6x} \left( 3 \cos3x - 4 \sin3x \right)\]
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