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Question #25d27

1 Answer

Mar 2, 2017

$(df_x)/(dx)(4)=color(green)(1/4)$

Explanation:

According to the limit definition, for $f(x)=sqrt(x)$
$(df_x)/(dx)=lim_(hrarr0) (f(x+h)-f(x))/h$

$color(white)("XXX")=lim_(hrarr0)(sqrt(x+h)-sqrt(x))/h$

$color(white)("XXX")=lim_(hrarr0)((sqrt(x+h)-sqrt(x)))/h * ((sqrt(x+h)+sqrt(x)))/((sqrt(x+h)+sqrt(x)))$

$color(white)("XXX")=lim_(hrarr0)(x+h-x)/(h * (sqrt(x+h)+sqrt(x)))$

$color(white)("XXX")=lim_(hrarr0) cancel(h)/(cancel(h) * (sqrt(x+h)+sqrt(x)))$

$color(white)("XXX")=lim_(hrarr0)1/(sqrt(x+h)+sqrt(x))$

$color(white)("XXX")=1/(2sqrt(x))$

$color(white)("XXX")=1/2 * 1/sqrt(x)$

$color(white)("XXX")=1/2 x^(-1/2)$

At $x=4$
$color(white)("XXX")(df_x)/(dx)(4) = 1/2 xx 4^(-1/2)$

$color(white)("XXXXXXX")=1/2 xx 1/2$

$color(white)("XXXXXXX")=1/4$

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