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Continuity and Differentiability — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsContinuity and Differentiability3 Marks
Question
Find the second-order derivative of the function exsin 5x
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Answer

Let y = exsin 5x

$\therefore \frac{{dy}}{{dx}} = {e^x}\frac{d}{{dx}}\sin 5x + \sin 5x\frac{d}{{dx}}{e^x}$

$= {e^x}\cos 5x\frac{d}{{dx}}5x + \sin 5x.{e^x} $ $= {e^x}\cos 5x \times 5 + {e^x}\sin 5x$

= ex(5 cos 5x + sin 5x)

$\Rightarrow \frac{{{d^2}y}}{{d{x^2}}} = {e^x}\frac{d}{{dx}}\left( {5\cos 5x + \sin 5x} \right) + \left( {5\cos 5x + \sin 5x} \right)\frac{d}{{dx}}{e^x}$

$= {e^x}\left[ {5\left( { - \sin 5x} \right) \times 5 + \left( {\cos 5x} \right) \times 5} \right] $ + (5 cos 5x + sin 5x)ex

= ex(-25 sin 5x + 5 cos 5x + 5 cos 5x + sin 5x)

= ex(10 cos 5x - 24 sin 5x)

= 2ex(5 cos 5x - 12 sin 5x)

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