# What is the derivative of sqrt(5x+sqrt(5x+sqrt(5x))) ?

Nov 13, 2017

$\frac{d}{\mathrm{dx}} \sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}$

$= \frac{1}{2 \sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}} \left(5 + \frac{1}{2 \sqrt{5 x + \sqrt{5 x}}} \left(5 + \frac{5}{2 \sqrt{5 x}}\right)\right)$

#### Explanation:

$\frac{d}{\mathrm{dx}} \sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}$

$= \frac{d}{\mathrm{dx}} {\left(5 x + {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}$

$= \frac{1}{2} {\left(5 x + {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \cdot \frac{d}{\mathrm{dx}} \left(5 x + {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)$

$= \frac{1}{2} {\left(5 x + {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \cdot \left(5 + \frac{1}{2} {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \cdot \frac{d}{\mathrm{dx}} \left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)\right)$

$= \frac{1}{2} {\left(5 x + {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \cdot \left(5 + \frac{1}{2} {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \cdot \left(5 + \frac{1}{2} {\left(5 x\right)}^{- \frac{1}{2}} \cdot \frac{d}{\mathrm{dx}} \left(5 x\right)\right)\right)$

$= \frac{1}{2} {\left(5 x + {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \cdot \left(5 + \frac{1}{2} {\left(5 x + {\left(5 x\right)}^{\frac{1}{2}}\right)}^{- \frac{1}{2}} \cdot \left(5 + \frac{1}{2} {\left(5 x\right)}^{- \frac{1}{2}} \cdot 5\right)\right)$

$= \frac{1}{2 \sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}} \left(5 + \frac{1}{2 \sqrt{5 x + \sqrt{5 x}}} \left(5 + \frac{5}{2 \sqrt{5 x}}\right)\right)$