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Differentiate From First Principles:

Hi! Me again :biggrin:

Anyways. Yeah. I'm differenting from first principles and I'm having trouble with certain functions!

I'll give a simple example, and then hopefully, with your help, I can master these particular functions.

Differentiate, from first principles

f(x) = \frac{1}{x}

I get:

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

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

What's to be done here then? Would I use the index laws to get rid of the fractions on top, as such:

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

Then multiply this by

\frac{(x+h)^{-1}+x^{-1}}{(x+h)^{-1}+x^{-1}}

f'(x) = \lim_{h \to 0} \frac{(x+h)^{-1}-x^{-1}}{h} \times \frac{(x+h)^{-1}+x^{-1}}{(x+h)^{-1}+x^{-1}}

f'(x) = \lim_{h \to 0} \frac{(x+h)^{-2}-x^{-2}}{h((x+h)^{-1}+x^{-1})}

When I multiply this out past this point I get ridiculous and unhelpful fractions! Any ideas?!

Reply 1

generalebriety

Try writing

\displaystyle \frac{1}{x+h} - \frac{1}{x}

as a single fraction over a common denominator.

Reply 2

Mush

generalebriety

Try writing

\displaystyle \frac{1}{x+h} - \frac{1}{x}

as a single fraction over a common denominator.

\frac{-h}{x(x+h)}

=\frac{-h}{x^{2}+hx}

Then what

Reply 3

figureeight

Try shoving that on the top of the fraction you had at the third line from the bottom, before you multiplied it by some crazy fraction that = 1 :s-smilie:

Reply 4

generalebriety

Mush

\frac{-h}{x(x+h)}

=\frac{-h}{x^{2}+hx}

Then what

What do you think? Put it back into the thing you were trying to work out. :s-smilie:

Reply 5

The Bachelor

Mush

\frac{1}{x+h}-\frac{1}{x}=\frac{-h}{x(x+h)}

Then what

Then

\lim_{h \rightarrow 0} \frac{\frac{1}{x+h}-\frac{1}{x}}{h}=\lim_{h \rightarrow 0} \frac{\frac{-h}{x(x+h)}}{h}

, does it not?

Reply 6

Mush

Haha silly me. Got it guys. Thanks a lot! REP IS IN ORDER :biggrin:

Reply 7

blue_n_white

Hi, ive got a problem with a question, can anyone help me out?

Differentiate from first principles:

f(x) = \sqrt{x^2-1}

ive tried:

\lim_{h\rightarrow0} \frac{(x^2+h^2+2xh-1)^{1/2}-(x^2-1)^{1/2}}{h}

but i'm now stuck!!

Reply 8

The Bachelor

Try multiplying by

\frac{\sqrt{(x+h)^2-1}+\sqrt{x^2-1}}{\sqrt{(x+h)^2-1}+\sqrt{x^2-1}}

cheers got it now

Could someone help me with this one? From first principles, find the derivative of 1/(sqrt(x^2-1))

Reply 11

TeeEm

Original post by tonka121

Could someone help me with this one? From first principles, find the derivative of 1/(sqrt(x^2-1))

do you know the basic principles?

Reply 12

tonka121

Yeah I know the basic principles

Reply 13

TenOfThem

Original post by tonka121

Could someone help me with this one? From first principles, find the derivative of 1/(sqrt(x^2-1))

Please do not resurrect old threads - it is against the rules

Reply 14

tonka121

Sorry - I hadn't actually seen that it was so old!

Reply 15

TeeEm

Original post by tonka121

Sorry - I hadn't actually seen that it was so old!

Please start a fresh thread

Please post any work you have done

please quote in your replies because I am working also and I need to see notifications in order to respond

(I do not understand why resurrecting a thread is a problem.
Why not lock them then?)