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unimatix
 one year ago
Derivatives without using shortcut method.
Problem:
g(x) = x sqrt(x)
unimatix
 one year ago
Derivatives without using shortcut method. Problem: g(x) = x sqrt(x)

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unimatix
 one year ago
Best ResponseYou've already chosen the best response.0Going to show what I have as of now: \[g(x) = xsqrt(x)\] \[y = xsqrt(x)\] \[y + \triangle y = x + \triangle(x)\sqrt{ (x+\triangle x ) )}\]

unimatix
 one year ago
Best ResponseYou've already chosen the best response.0dw:1441578438052:dw

triciaal
 one year ago
Best ResponseYou've already chosen the best response.0dw:1441579069139:dw

triciaal
 one year ago
Best ResponseYou've already chosen the best response.0dw:1441579234601:dw

triciaal
 one year ago
Best ResponseYou've already chosen the best response.0this is going too far back

unimatix
 one year ago
Best ResponseYou've already chosen the best response.0I know the product rule. I'm trying to do it without though.

unimatix
 one year ago
Best ResponseYou've already chosen the best response.0I tried to rationalize the numerator on the left and got nothing.

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0\[g(x) = x\sqrt{x}\]\[g(x+h) = (x+h)(\sqrt{x+h})\]\[\frac{g(x+h)g(x)}{h}=\frac{(x+h)(\sqrt{x+h})x\sqrt{x}}{h} \]

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0Meh.. the only thing that'll work here would be : \[=\frac{\sqrt{(x+h)^3}x^{3/2}}{h}= \frac{(x+h)^{3/2}x^{3/2}}{h} \]

amistre64
 one year ago
Best ResponseYou've already chosen the best response.2you are trying to work first principles .. the limit definition of a derivative right?

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0Mmhmm... \(\lim_{h\rightarrow 0}\)

amistre64
 one year ago
Best ResponseYou've already chosen the best response.2(x+h) sqrt(x+h)  x sqrt(x)  h x sqrt(x+h) +hsqrt(x+h)  x sqrt(x)  h x (sqrt(x+h)  sqrt(x)) sqrt(x+h) +  h x (h) sqrt(x+h) +  h(sqrt(x+h) + sqrt(x)) x sqrt(x+h) +  sqrt(x+h) + sqrt(x))

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0Not the third step.

amistre64
 one year ago
Best ResponseYou've already chosen the best response.2(ac+b)/a = c + (b/a)

amistre64
 one year ago
Best ResponseYou've already chosen the best response.2you just split the fraction into convenient terms, and the left side is just a simplified result of h/h sqrt(x+h)

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0Oh, I see what you did now.

amistre64
 one year ago
Best ResponseYou've already chosen the best response.2\[\frac{a+bh+c}{h}\color{red}{\implies}\frac{bh}{h}+\frac{a+c}{h}\]

amistre64
 one year ago
Best ResponseYou've already chosen the best response.2the rest is taken care of by a conjugate

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0Just trying to solve it by myself real quick and crosschecking between my work and your method :)

Jhannybean
 one year ago
Best ResponseYou've already chosen the best response.0Then.... \[\lim_{h\rightarrow 0} \left(\sqrt{x+h} +\frac{x}{2\sqrt{x}}\right) \]\[=\sqrt{x}+\frac{x}{2\sqrt{x}}\]\[=\frac{2x+x}{2\sqrt{x}}\]\[=\frac{3x}{2\sqrt{x}}\]\[=\frac{3x^{3/2}}{2x}\]\[=\boxed{\frac{3}{2}x^{1/2}}\]
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