Question

Find the derivative of the given function at the indicated point.

f(x)=1/x,a = 2

f′(a)= lim (f(a+h) - f(a))/h
h --> 0

Answer 1

Could you show steps please, im really confused.

Answer 2

rewrite your function in index form

\[f(x) = x^{-1}\]

can you differentiate the function..?

Answer 3

You have that \(f(x)=1/x\) and are asked to compute its derivative at \(a=2\). Then\[f'(a)=\lim_{h\to0}\frac{1/(2+h)-1/2}{h}.\]Can you simplify this and compute its limit?

Answer 4

Thanks for getting me started across. Ill see if I can simplify.

Answer 5

oops 1st principles


\[\lim_{h \rightarrow0} \frac{ \frac{1}{x + h} - \frac{1}{h}}{h}\]

put the fractions in the numerator over a common denominator

\[\lim_{h \rightarrow 0} \frac{\frac{x - (x +h)}{x(x+h)}}{h}\]

so it can then be simplified to

\[\lim_{h \rightarrow 0} \frac{\frac{-h}{x(x + h)}}{h}\]

or

\[\lim_{h \rightarrow 0} \frac{-h}{hx(x + h)}\]

cancel the common factor and then substitute h = 0

to get the derivative

Answer 6

Oh I see, would it also simplify further to 1/x^2.
Also at a=2, the deravitive of f(x) = 1/4 right?

Answer 7

well it simplifies to -1/x^2

Answer 8

so you need to check you value for a = 2

Answer 9

Oh, careless mistake. Thank you so much, this really helped.