What is the derivative of $\sqrt{2 x}$ $sqrt(2x)$ ?

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What is the derivative of $\sqrt{2 x}$ $sqrt(2x)$ ?

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Answer 1 · 2017-03-26T12:18:07

$\frac{d}{d x} \sqrt{2 x} = \frac{\sqrt{2}}{2 \sqrt{x}}$ $d/(dx) sqrt(2x) = sqrt(2)/(2sqrt(x))$

Explanation:

For any non-zero power $p$ $p$ of $x$ $x$ we have:

$\frac{d}{d x} x^{p} = p x^{p - 1}$ $d/(dx) x^p = p x^(p-1)$

So:

$\frac{d}{d x} \sqrt{2 x} = \frac{d}{d x} \sqrt{2} \sqrt{x}$ $d/(dx) sqrt(2x) = d/(dx) sqrt(2)sqrt(x)$

$\frac{d}{d x} \sqrt{2 x} = \frac{d}{d x} \sqrt{2} x^{\frac{1}{2}}$ $color(white)(d/(dx) sqrt(2x)) = d/(dx) sqrt(2)x^(1/2)$

$\frac{d}{d x} \sqrt{2 x} = \sqrt{2} \cdot \frac{1}{2} x^{- \frac{1}{2}}$ $color(white)(d/(dx) sqrt(2x)) = sqrt(2)*1/2x^(-1/2)$

$\frac{d}{d x} \sqrt{2 x} = \frac{\sqrt{2}}{2} \cdot \frac{1}{x^{\frac{1}{2}}}$ $color(white)(d/(dx) sqrt(2x)) = sqrt(2)/2*1/x^(1/2)$

$\frac{d}{d x} \sqrt{2 x} = \frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{x}}$ $color(white)(d/(dx) sqrt(2x)) = sqrt(2)/2*1/sqrt(x)$

$\frac{d}{d x} \sqrt{2 x} = \frac{\sqrt{2}}{2 \sqrt{x}}$ $color(white)(d/(dx) sqrt(2x)) = sqrt(2)/(2sqrt(x))$

Answer 2 · 2017-03-26T12:28:07

$\frac{d}{d x} \sqrt{2 x} = \frac{\sqrt{2}}{2 \sqrt{x}}$ $d/(dx) sqrt(2x) = sqrt(2)/(2sqrt(x))$

Explanation:

Here is how to find it using the limit definition of derivative...

$d/(dx) sqrt(2x) = lim_(h->0) (sqrt(2(x+h))-sqrt(2x))/h$

$color(white)(d/(dx) sqrt(2x)) = lim_(h->0) ((sqrt(2(x+h))-sqrt(2x))(sqrt(2(x+h))+sqrt(2x)))/(h(sqrt(2(x+h))+sqrt(2x)))$

$color(white)(d/(dx) sqrt(2x)) = lim_(h->0) ((sqrt(2(x+h)))^2-(sqrt(2x))^2)/(h(sqrt(2(x+h))+sqrt(2x)))$

$color(white)(d/(dx) sqrt(2x)) = lim_(h->0) (2(x+h)-2x)/(h(sqrt(2(x+h))+sqrt(2x)))$

$color(white)(d/(dx) sqrt(2x)) = lim_(h->0) (2color(red)(cancel(color(black)(h))))/(color(red)(cancel(color(black)(h)))(sqrt(2(x+h))+sqrt(2x)))$

$color(white)(d/(dx) sqrt(2x)) = lim_(h->0) 2/(sqrt(2(x+h))+sqrt(2x))$

$color(white)(d/(dx) sqrt(2x)) = 2/(sqrt(2(x+0))+sqrt(2x))$

$color(white)(d/(dx) sqrt(2x)) = 1/(sqrt(2x))$

$color(white)(d/(dx) sqrt(2x)) = sqrt(2)/(2sqrt(x))$

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What is the derivative of $\sqrt{2 x}$ $sqrt(2x)$ ?

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