Can someone please help me, I don't know what I am doing....

[maxboy0801]

Can someone please help me, I don't know what I am doing....

Problem: Find [f(x + h) - f(x)]/h if f(x) = x + ln x

 

[soroban]

Re: Can someone please help me, I don't know what I am doing

Hello, maxboy0801!

Find \(\displaystyle \frac{f(x + h)\,-\,f(x)}{h}\) .if \(\displaystyle f(x)\:=\:x\,+\,\ln(x)\)

I always break up the Difference Quotient into three steps:
. . (1) Find \(\displaystyle f(x+h)\) . . . replace \(\displaystyle x\) with \(\displaystyle x+h.\) .(Simplify if possible)
. . (2) Subtract \(\displaystyle f(x)\) . . .subtract the original function. .(Simplify if possible)
. . (3) Divide by \(\displaystyle h.\) .(Simplify if possible)

\(\displaystyle (1)\;\;f(x+h)\:=\x+h)\,+\,\ln(x+h)\)


\(\displaystyle (2)\;\;f(x+h)\,-\,f(x)\:=\:[(x+h)\,+\,\ln(x+h)]\,-\,[x + \ln(x)]\)

. . . . . . . . . . . . . . . . . .\(\displaystyle =\:x\,+\,h\,+\,\ln(x+h)\,-\,x\,-\,\ln(x)\)

. . . . . . . . . . . . . . . . . .\(\displaystyle =\:h\,+\,\ln(x+h)\,-\,\ln(x)\)

. . . . . . . . . . . . . . . . . .\(\displaystyle =\:h\,+\,\ln\left(\frac{x+h}{x}\right)\)


\(\displaystyle (3)\;\;\frac{f(x+h)\,-\,f(x)}{h}\;=\;\frac{h\,+\,\ln\left(\frac{x+h}{x}\right)}{h}\)

. . . . . . . . . . . . . . . . \(\displaystyle =\:\frac{h}{h}\,+\,\frac{\ln\left(\frac{x+h}{x}\right)}{h}\)

. . . . . . . . . . . . . . . . \(\displaystyle =\:1\,+\,\frac{1}{h}\ln\left(\frac{x+h}{x}\right)\)