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Using the limit definition, how do you find the derivative of #f(x) = 2sqrtx#?

Answer 1

Christopher Allers

#f'(x) = 1/sqrt(x)#

Using the limit definition of a derivative, we have

#f'(x) = lim_(h->0) (f(x+h)-f(x))/h#

# = lim_(h->0)(2sqrt(x+h)-2sqrt(x))/h#

#= 2lim_(h->0)(sqrt(x+h)-sqrt(x))/h*(sqrt(x+h)+sqrt(x))/(sqrt(x+h)+sqrt(x))#

#=2lim_(h->0)(x+h-x)/(h(sqrt(x+h)+sqrt(x))#

#=2lim_(h->0) 1/(sqrt(x+h)+sqrt(x))#

#= 2*1/(sqrt(x+0)+sqrt(x))#

#=2*1/(2sqrt(x))#

#=1/sqrt(x)#

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Answer 2

Charles Anctil

To find the derivative of ( f(x) = 2\sqrt{x} ) using the limit definition, follow these steps:

Start with the function ( f(x) = 2\sqrt{x} ).
Let ( h ) be a small change in ( x ).
Find ( f(x + h) ) by substituting ( x + h ) into the function: ( f(x + h) = 2\sqrt{x + h} ).
Use the difference quotient to find the slope of the secant line between ( x ) and ( x + h ):

[ \frac{f(x + h) - f(x)}{h} = \frac{2\sqrt{x + h} - 2\sqrt{x}}{h} ]

Simplify the expression.
Take the limit as ( h ) approaches 0:

[ \lim_{{h \to 0}} \frac{2\sqrt{x + h} - 2\sqrt{x}}{h} ]

Evaluate the limit to find the derivative.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.