# What is the second derivative of f(x)= sin(sqrt(3x-7))?

Jan 10, 2016

f''(x)=(-9(sqrt(3x-7)sin(sqrt(3x-7))+cos(sqrt(3x-7))))/(4(3x-7)^(3/2)

#### Explanation:

Finding the first derivative:

Use the chain rule (many times).

First Issue: the sine function. $\frac{d}{\mathrm{dx}} \left[\sin \left(u\right)\right] = \cos \left(u\right) \cdot u '$.

$f ' \left(x\right) = \cos \left(\sqrt{3 x - 7}\right) \cdot \frac{d}{\mathrm{dx}} \left[\sqrt{3 x - 7}\right]$

Second Issue: the square root. $\frac{d}{\mathrm{dx}} \left[\sqrt{u}\right] = \frac{d}{\mathrm{dx}} \left[{u}^{\frac{1}{2}}\right] = \frac{1}{2} {u}^{- \frac{1}{2}} \cdot u ' = \frac{u '}{2 \sqrt{u}}$

f'(x)=cos(sqrt(3x-7))*3/(2sqrt(3x-7))=color(blue)((3cos(sqrt(3x-7)))/(2sqrt(3x-7))

Finding the second derivative:

Use the quotient rule (and more chain rule).

$f ' ' \left(x\right) = \frac{2 \sqrt{3 x - 7} \textcolor{g r e e n}{\frac{d}{\mathrm{dx}} \left[3 \cos \left(\sqrt{3 x - 7}\right)\right]} - 3 \cos \left(\sqrt{3 x - 7}\right) \textcolor{red}{\frac{d}{\mathrm{dx}} \left[2 \sqrt{3 x - 7}\right]}}{2 \sqrt{3 x - 7}} ^ 2$

Find each internal derivative separately:

$\frac{d}{\mathrm{dx}} \left[3 \cos \left(\sqrt{3 x - 7}\right)\right]$

This will be almost identical to finding $\frac{d}{\mathrm{dx}} \left[\sqrt{3 x - 7}\right]$, except that it will turn into a negative sine function and have the $3$ multiplied by it. We could do the work, but why redo something we've already essentially done?

color(green)(d/dx[3cos(sqrt(3x-7))]=(-9sin(sqrt(3x-7)))/(2sqrt(3x-7))

The other derivative is

$\frac{d}{\mathrm{dx}} \left[2 \sqrt{3 x - 7}\right]$

This, again, will be twice what we determined earlier, since we've basically already done this differentiation.

color(red)(d/dx[2sqrt(3x-7)]=3/sqrt(3x-7)

Plug these back in.

$f ' ' \left(x\right) = \frac{\frac{2 \sqrt{3 x - 7} \left(- 9 \sin \left(\sqrt{3 x - 7}\right)\right)}{2 \sqrt{3 x - 7}} - \frac{9 \cos \left(\sqrt{3 x - 7}\right)}{\sqrt{3 x - 7}}}{4 \left(3 x - 7\right)}$

Further simplification yields:

f''(x)=(-9(sqrt(3x-7)sin(sqrt(3x-7))+cos(sqrt(3x-7))))/(4(3x-7)^(3/2)