Question

if f(x)=sqrt(x+5), then f(x+h)-f(x)/h = A/sqrt(x+5)+sqrt(Bx+Ch+D)
Find A, B, C, D

Answer 1

\[f(x)=\sqrt(x+5), then \frac{ f(x+h)-f(x) }{ h }= \frac{ A }{ \sqrt(x+5)+\sqrt(Bx+Ch+D) }\]

Answer 2

\[f(x) = \sqrt{x+5}\]
\[f(x+h) = \sqrt{(x+h)+5}\]
\[\frac{f(x+h) - f(x) }{h}= \frac{\sqrt{x+h+5}-\sqrt{x+5}}{h}\]Multiply numerator and denominator by the conjugate of the numerator
\[\frac{(\sqrt{x+h+5}-\sqrt{x+5})*(\sqrt{x+h+5}+\sqrt{x+5})}{h(\sqrt{x+h+5}+\sqrt{h+5})}\]The top is just the difference of squares, so
\[\frac{(x+h+5)-(x+5)}{h(\sqrt{x+h+5}+\sqrt{h+5})} = \frac{h}{h(\sqrt{x+h+5}+\sqrt{h+5})}\]
Can you take it from there?

By the way, multiplying by the conjugate in this fashion is a good technique to remember, as it shows up in many places.

Answer 3

yes, thank you very much