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$Find the second order derivatives of the functions given in Exercises 1 to 10.$

Q1. $y = x^{2} + 3 x + 2$

Solution

$\frac{d y}{d x} = 2 x + 3$ $\frac{d^{2} y}{d x^{2}} = 2$

Q2. $y = x^{20}$

Solution

$\frac{d y}{d x} = 20 x^{19}$ $\frac{d^{2} y}{d x^{2}} = 380 x^{18}$

Q3. $y = x \cos x$

Solution

Using product rule:

$\frac{d y}{d x} = \cos x - x \sin x$

Now differentiate again:

$\frac{d^{2} y}{d x^{2}} = - \sin x - (\sin x + x \cos x)$ $\frac{d^{2} y}{d x^{2}} = - 2 \sin x - x \cos x$

Q4. $y = \log x$

Solution

$\frac{d y}{d x} = \frac{1}{x}$ $\frac{d^{2} y}{d x^{2}} = - \frac{1}{x^{2}}$

Q5. $y = x^{3} \log x$

Solution

$\frac{d y}{d x} = 3 x^{2} \log x + x^{2}$ $\frac{d^{2} y}{d x^{2}} = 6 x \log x + 5 x$

Q6. $y = e^{x} \sin 5 x$

Solution

$\frac{d y}{d x} = e^{x} (\sin 5 x + 5 \cos 5 x)$ $\frac{d^{2} y}{d x^{2}} = e^{x} (- 24 \sin 5 x + 10 \cos 5 x)$

Q7. $y = e^{6 x} \cos 3 x$

Solution $\frac{d y}{d x} = e^{6 x} (6 \cos 3 x - 3 \sin 3 x)$ $\frac{d^{2} y}{d x^{2}} = e^{6 x} (27 \cos 3 x - 36 \sin 3 x)$

Q8. $y = \tan^{- 1} x$

Solution $\frac{d y}{d x} = \frac{1}{1 + x^{2}}$ $\frac{d^{2} y}{d x^{2}} = - \frac{2 x}{(1 + x^{2})^{2}}$

Q9. $y = \log (\log x)$

Solution $\frac{d y}{d x} = \frac{1}{x \log x}$ $\frac{d^{2} y}{d x^{2}} = - \frac{\log x + 1}{x^{2} (\log x)^{2}}$

Q10. $y = \sin (\log x)$

Solution $\frac{d y}{d x} = \frac{\cos (\log x)}{x}$ $\frac{d^{2} y}{d x^{2}} = \frac{- \cos (\log x) - \sin (\log x)}{x^{2}}$

Q11. If $y = 5 \cos x - 3 \sin x$ , show that $\frac{d^{2} y}{d x^{2}} + y = 0$

Solution

$\frac{d y}{d x} = - 5 \sin x - 3 \cos x$ $\frac{d^{2} y}{d x^{2}} = - 5 \cos x + 3 \sin x$

Now, $\frac{d^{2} y}{d x^{2}} + y = (- 5 \cos x + 3 \sin x) + (5 \cos x - 3 \sin x)$

$\frac{d^{2} y}{d x^{2}} + y = 0$

$Hence proved$

proteacher.in

Find the second order derivatives of the functions given in Exercises 1 to 10.

Q1. y=x2+3x+2

Solution

Q2. y=x20

Solution

Q3. y=xcos⁡x

Solution

Q4. y=log⁡x

Solution

Q5. y=x3log⁡x

Solution

Q6. y=exsin⁡5x

Solution

Q7. y=e6xcos⁡3x

Solutiondydx=e6x(6cos⁡3x−3sin⁡3x)d2ydx2=e6x(27cos⁡3x−36sin⁡3x)

Q8. y=tan⁡−1x

Solutiondydx=11+x2d2ydx2=−2x(1+x2)2

Q9. y=log⁡(log⁡x)

Solutiondydx=1xlog⁡xd2ydx2=−log⁡x+1x2(log⁡x)2

Q10. y=sin⁡(log⁡x)y=sin(logx)

Solutiondydx=cos⁡(log⁡x)xd2ydx2=−cos⁡(log⁡x)−sin⁡(log⁡x)x2

Q11. If y=5cos⁡x−3sin⁡x, show that d2ydx2+y=0

Solution

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$Find the second order derivatives of the functions given in Exercises 1 to 10.$

Q1. $y = x^{2} + 3 x + 2$

Q2. $y = x^{20}$

Q3. $y = x \cos x$

Q4. $y = \log x$

Q5. $y = x^{3} \log x$

Q6. $y = e^{x} \sin 5 x$

Q7. $y = e^{6 x} \cos 3 x$

Solution $\frac{d y}{d x} = e^{6 x} (6 \cos 3 x - 3 \sin 3 x)$ $\frac{d^{2} y}{d x^{2}} = e^{6 x} (27 \cos 3 x - 36 \sin 3 x)$

Q8. $y = \tan^{- 1} x$

Solution $\frac{d y}{d x} = \frac{1}{1 + x^{2}}$ $\frac{d^{2} y}{d x^{2}} = - \frac{2 x}{(1 + x^{2})^{2}}$

Q9. $y = \log (\log x)$

Solution $\frac{d y}{d x} = \frac{1}{x \log x}$ $\frac{d^{2} y}{d x^{2}} = - \frac{\log x + 1}{x^{2} (\log x)^{2}}$

Q10. $y = \sin (\log x)$

Solution $\frac{d y}{d x} = \frac{\cos (\log x)}{x}$ $\frac{d^{2} y}{d x^{2}} = \frac{- \cos (\log x) - \sin (\log x)}{x^{2}}$

Q11. If $y = 5 \cos x - 3 \sin x$ , show that $\frac{d^{2} y}{d x^{2}} + y = 0$

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