How do you find the derivative of f(x)=x using the formal definition?

f′(a)=limh→0(f(a+h)−f(a)h)=limh→0(a+h−ah)=limh→0((a+h−a)(a+h+a)h(a+h+a))=limh→0((a+h)-ah(a+h+a))=limh→0(1a+h+a)=12aThat is:f′(x)=12x

Best solutions to - "How do you find the derivative of f(x)=x..."

Max Williams

Answered question

2022-04-15

How do you find the derivative of

f (x) = \sqrt{x}

using the formal definition?

Answer & Explanation

Naima Silva

Beginner2022-04-16Added 9 answers

From the definition of derivative:

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

I would try to rationalize the numerator (although it seems strange...).

= lim_{h \to 0} (\frac{\sqrt{x + h} - \sqrt{x}}{h}) \cdot \frac{(\sqrt{x + h} + \sqrt{x})}{(\sqrt{x + h} + \sqrt{x})} =

= lim_{h \to 0} \frac{x + h - x}{h \cdot (\sqrt{x + h} + \sqrt{x})} =

= lim_{h \to 0} \frac{h}{h \cdot (\sqrt{x + h} + \sqrt{x})} =

= lim_{h \to 0} \frac{1}{(\sqrt{x + h} + \sqrt{x})} =

h \to 0

= \frac{1}{2 \sqrt{x}}

fallendreamsit2p

Beginner2022-04-17Added 14 answers

$f^{'} (a) = lim_{h \to 0} (\frac{f (a + h) - f (a)}{h})$
$= lim_{h \to 0} (\frac{\sqrt{a + h} - \sqrt{a}}{h})$
$= lim_{h \to 0} (\frac{(\sqrt{a + h} - \sqrt{a}) (\sqrt{a + h} + \sqrt{a})}{h (\sqrt{a + h} + \sqrt{a})})$
$= \lim_{h \to 0} (\frac{(a + h) - a}{h (\sqrt{a + h} + \sqrt{a})})$
$= lim_{h \to 0} (\frac{1}{\sqrt{a + h} + \sqrt{a}})$
$= \frac{1}{2 \sqrt{a}}$
That is:
$f^{'} (x) = \frac{1}{2 \sqrt{x}}$

Do you have a similar question?

Recalculate according to your conditions!

New Questions in Calculus 1

Didn't find what you were looking for?