Hello there,
I am self-studying calculus using The Complete Idiot's Guide to Calculus, and there is a problem asking me to calculate the derivative of a function with the alternate difference quotient.
However, for extra practise, I am trying to find the derivative of a function with the regular difference quotient. I am having trouble on one step so I would appreciate any help.
Thank you.
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1. Calculate the derivative \(\displaystyle f(x) = \sqrt{x + 1}\).
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\(\displaystyle \lim_{\Delta x \to 0}\frac {f(x + \Delta x) - f(x)}{\Delta x}\)
Determining \(\displaystyle f(x + \Delta x)\):
\(\displaystyle f(x + \Delta x) = \sqrt{(x + \Delta x) + 1}\)
Therefore:
\(\displaystyle \lim_{\Delta x \to 0}\frac {f(x + \Delta x) - f(x)}{\Delta x}\)
= \(\displaystyle \lim_{\Delta x \to 0}\frac {\sqrt{x + \Delta x + 1} - \sqrt{x + 1}}{\Delta x}\)
How would I eliminate the square roots to proceed with the algebra?
I am self-studying calculus using The Complete Idiot's Guide to Calculus, and there is a problem asking me to calculate the derivative of a function with the alternate difference quotient.
However, for extra practise, I am trying to find the derivative of a function with the regular difference quotient. I am having trouble on one step so I would appreciate any help.
Thank you.
---
1. Calculate the derivative \(\displaystyle f(x) = \sqrt{x + 1}\).
---
\(\displaystyle \lim_{\Delta x \to 0}\frac {f(x + \Delta x) - f(x)}{\Delta x}\)
Determining \(\displaystyle f(x + \Delta x)\):
\(\displaystyle f(x + \Delta x) = \sqrt{(x + \Delta x) + 1}\)
Therefore:
\(\displaystyle \lim_{\Delta x \to 0}\frac {f(x + \Delta x) - f(x)}{\Delta x}\)
= \(\displaystyle \lim_{\Delta x \to 0}\frac {\sqrt{x + \Delta x + 1} - \sqrt{x + 1}}{\Delta x}\)
How would I eliminate the square roots to proceed with the algebra?
