How do you use the limit definition to find the derivative of $2 \sqrt{x} - \frac{1}{2 \sqrt{x}}$ $2sqrtx-1/(2sqrtx)$ ?

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Answer 1 · 2017-10-14T18:08:36

Using the lit definition we have:

$d/dx(2sqrtx-1/(2sqrtx)) = lim_(h->0) (2sqrt(x+h) -1/(2sqrt(x+h)) - 2sqrtx+1/(2sqrtx))/h$

$d/dx(2sqrtx-1/(2sqrtx)) = lim_(h->0) (2sqrt(x+h) - 2sqrtx) /h - (1/(2sqrt(x+h))-1/(2sqrtx))/h$

Now rationalize the numerator of the first term:

$(2sqrt(x+h) - 2sqrtx) /h = (2sqrt(x+h) - 2sqrtx) /h xx (sqrt(x+h) + sqrtx)/ (sqrt(x+h) +sqrtx)$

$(2sqrt(x+h) - 2sqrtx) /h = (2(x+h) - 2x) /(h (sqrt(x+h) +sqrtx)$

$(2sqrt(x+h) - 2sqrtx) /h = (2cancelh) /(cancelh (sqrt(x+h) +sqrtx)$

So:

$lim_(h->0) (2sqrt(x+h) - 2sqrtx) /h = 1/sqrtx$

For the second term of the sum:

$(1/(2sqrt(x+h))-1/(2sqrtx))/h = (2sqrtx -2sqrt(x+h))/(hsqrtxsqrt(x+h))$

and in the same way as above:

$(1/(2sqrt(x+h))-1/(2sqrtx))/h = (2x -2(x+h))/(hsqrtxsqrt(x+h)(sqrtx +sqrt(x+h))$

$(1/(2sqrt(x+h))-1/(2sqrtx))/h = (-2cancelh)/(cancelhsqrtxsqrt(x+h)(sqrtx +sqrt(x+h))$

so that:

$lim_(h->0)(1/(2sqrt(x+h))-1/(2sqrtx))/h = -1/(xsqrtx)$

Finally:

$d/dx(2sqrtx-1/(2sqrtx)) = 1/sqrtx+1/(xsqrtx)$

How do you use the limit definition to find the derivative of 2sqrtx-1/(2sqrtx)2√x−12√x2sqrtx-1/(2sqrtx)?