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Good afternoon can anyone help me out how to solve this kind of problem about derivatives thank you

Gwapako

New member
Jul 20, 2020
3
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skeeter

Well-known member
MHB Math Helper
Mar 1, 2012
702
I assume you’re referring to a form of the following limit ...

$\displaystyle f’(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x}$

for ease in typing it out, let $\Delta x = h$

$\displaystyle y’ = \lim_{h \to 0} \dfrac{1}{h} \left[\dfrac{2(x+h)-1}{2(x+h)+1} - \dfrac{2x-1}{2x+1} \right]$

from here, it’s just the algebra drill of combining the two fractions using a common denominator and ultimately getting the $h$ in the leading $\dfrac{1}{h}$ factor to divide out with an $h$ factor in the numerator.


by the same token, let $\Delta t = h$ in the second limit; also, using the properties of limits will make the overall task a bit easier ...

$\displaystyle x’ = 3 \cdot \lim_{h \to 0} \dfrac{(t+h)^2-t^2}{h} - 2 \cdot \lim_{h \to 0} \dfrac{\sqrt{t+h} - \sqrt{t}}{h}$

the limit of the first term is straightforward; the second will require you to rationalize the numerator