How do you differentiate f(x)=#1/sqrt(x-4)# using first principles?

Question

Elijah Abram · Accepted Answer

#f(x)=1/sqrt(x-4)# #f'(x)=lim_(hrarr0) (f(x+h)-f(x))/h# # = lim_(hrarr0) (1/sqrt((x-4)+h)-1/sqrt((x-4)))/h# # = lim_(hrarr0) ((sqrt(x-4)-sqrt((x-4)+h))/(sqrt((x-4)+h)sqrt((x-4))))/(h/1)# # = lim_(hrarr0) (sqrt(x-4)-sqrt((x-4)+h))/(sqrt((x-4)+h)sqrt((x-4))) * 1/h# # = lim_(hrarr0) ((sqrt(x-4)-sqrt((x-4)+h)))/(hsqrt((x-4)+h)sqrt((x-4))) * ((sqrt(x-4)+sqrt((x-4)+h)))/((sqrt(x-4)+sqrt((x-4)+h)))# # = lim_(hrarr0) ((x-4)-((x-4)+h))/(hsqrt((x-4)+h)sqrt((x-4))(sqrt(x-4)+sqrt((x-4)+h)))# # = lim_(hrarr0) (-cancel(h)^1)/(cancel(h)sqrt((x-4)+h)sqrt((x-4))(sqrt(x-4)+sqrt((x-4)+h)))# # = lim_(hrarr0) (-1)/(sqrt((x-4)+h)sqrt((x-4))(sqrt(x-4)+sqrt((x-4)+h)))# # =  (-1)/(sqrt((x-4)+0)sqrt((x-4))(sqrt(x-4)+sqrt((x-4)+0)))# # =  (-1)/((x-4)(2sqrt(x-4))) =(-1)/(2(x-4)sqrt(x-4)) = (-1)/(2(x-4)^(3/2))#

Emily Ammerman · Answer

undefined To differentiate $ f(x) = \frac{1}{\sqrt{x - 4}} $ using first principles, you need to apply the definition of the derivative:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

1. Substitute $ f(x) = \frac{1}{\sqrt{x - 4}} $ into the formula.
2. Expand $ f(x + h) $ and $ f(x) $.
3. Simplify the expression.
4. Take the limit as $ h $ approaches 0.

$$ f(x + h) = \frac{1}{\sqrt{x + h - 4}}, $$
$$ f(x) = \frac{1}{\sqrt{x - 4}}. $$

$$ f'(x) = \lim_{h \to 0} \frac{\frac{1}{\sqrt{x + h - 4}} - \frac{1}{\sqrt{x - 4}}}{h} $$

$$ = \lim_{h \to 0} \frac{\sqrt{x - 4} - \sqrt{x + h - 4}}{h \sqrt{x - 4}\sqrt{x + h - 4}} $$

$$ = \lim_{h \to 0} \frac{\sqrt{x - 4} - \sqrt{x + h - 4}}{h} \cdot \frac{\sqrt{x - 4} + \sqrt{x + h - 4}}{\sqrt{x - 4}\sqrt{x + h - 4}} $$

$$ = \lim_{h \to 0} \frac{x - 4 - (x + h - 4)}{h(\sqrt{x - 4} + \sqrt{x + h - 4})\sqrt{x - 4}\sqrt{x + h - 4}} $$

$$ = \lim_{h \to 0} \frac{-h}{h(\sqrt{x - 4} + \sqrt{x + h - 4})\sqrt{x - 4}\sqrt{x + h - 4}} $$

$$ = \lim_{h \to 0} \frac{-1}{(\sqrt{x - 4} + \sqrt{x + h - 4})\sqrt{x - 4}\sqrt{x + h - 4}} $$

$$ = \frac{-1}{2\sqrt{x - 4}\sqrt{x - 4}} $$

$$ = \frac{-1}{2(x - 4)} $$