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Gwapako
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MHB Help with Derivatives Problem - Get Answers Now
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The discussion focuses on solving derivative problems using the limit definition of a derivative. The limit is expressed as f’(x) = lim(Δx→0) [f(x+Δx) - f(x)]/Δx, simplifying the notation by substituting Δx with h. The algebra involves combining fractions with a common denominator and canceling out h factors. Additionally, the second limit involves a straightforward limit and a more complex one that requires rationalizing the numerator. The conversation emphasizes the importance of algebraic manipulation and limit properties in finding derivatives.
- #2
skeeter
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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
$\displaystyle f’(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x}$
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
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