What is the derivative of $\sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}$ $sqrt(5x+sqrt(5x+sqrt(5x)))$ ?

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What is the derivative of $\sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}$ $sqrt(5x+sqrt(5x+sqrt(5x)))$ ?

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Answer 1 · 2017-11-13T22:46:31

$\frac{d}{d x} \sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}$ $d/(dx) sqrt(5x+sqrt(5x+sqrt(5x)))$

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

Explanation:

$\frac{d}{d x} \sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}$ $d/(dx) sqrt(5x+sqrt(5x+sqrt(5x)))$

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

$= \frac{1}{2} {(5 x + {(5 x + {(5 x)}^{\frac{1}{2}})}^{\frac{1}{2}})}^{- \frac{1}{2}} \cdot \frac{d}{d x} (5 x + {(5 x + {(5 x)}^{\frac{1}{2}})}^{\frac{1}{2}})$ $= 1/2 (5x+(5x+(5x)^(1/2))^(1/2))^(-1/2) * d/(dx) (5x+(5x+(5x)^(1/2))^(1/2))$

$= \frac{1}{2} {(5 x + {(5 x + {(5 x)}^{\frac{1}{2}})}^{\frac{1}{2}})}^{- \frac{1}{2}} \cdot (5 + \frac{1}{2} {(5 x + {(5 x)}^{\frac{1}{2}})}^{- \frac{1}{2}} \cdot \frac{d}{d x} (5 x + {(5 x)}^{\frac{1}{2}}))$ $= 1/2 (5x+(5x+(5x)^(1/2))^(1/2))^(-1/2) * (5+1/2 (5x+(5x)^(1/2))^(-1/2) * d/(dx) (5x+(5x)^(1/2)))$

$= \frac{1}{2} {(5 x + {(5 x + {(5 x)}^{\frac{1}{2}})}^{\frac{1}{2}})}^{- \frac{1}{2}} \cdot (5 + \frac{1}{2} {(5 x + {(5 x)}^{\frac{1}{2}})}^{- \frac{1}{2}} \cdot (5 + \frac{1}{2} {(5 x)}^{- \frac{1}{2}} \cdot \frac{d}{d x} (5 x)))$ $= 1/2 (5x+(5x+(5x)^(1/2))^(1/2))^(-1/2) * (5+1/2 (5x+(5x)^(1/2))^(-1/2) * (5+ 1/2 (5x)^(-1/2) * d/(dx) (5x)))$

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

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

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What is the derivative of $\sqrt{5 x + \sqrt{5 x + \sqrt{5 x}}}$ $sqrt(5x+sqrt(5x+sqrt(5x)))$ ?

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Explanation:

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