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How do you use the limit definition of the derivative to find the derivative of #f(x)=sqrtx#?

Answer 1

Ezra Adams

# f'(x)=1/(2sqrtx)#.

By defn., #f'(x)=lim_(trarrx) (f(t)-f(x))/(t-x)#.

For #f(x)=sqrtx=x^(1/2), f'(x)=lim_(trarrx) (t^(1/2)-x^(1/2))/(t-x)#

#=lim_(trarrx) (t^(1/2)-x^(1/2))/{(t^(1/2))^2-(x^(1/2))^2}#

#=lim_(trarrx) cancel((t^(1/2)-x^(1/2)))/{(t^(1/2)+x^(1/2))cancel((t^(1/2)-x^(1/2))}#

#=lim_(trarrx) 1/(t^(1/2)+x^(1/2)#

#=1/(x^(1/2)+x^(1/2))#

#=1/(2x^(1/2))#

#:. f'(x)=1/(2sqrtx)#.

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Answer 2

Isabella Ackerman

To find the derivative of ( f(x) = \sqrt{x} ) using the limit definition of the derivative:

Start with the function ( f(x) = \sqrt{x} ).
Use the definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ).
Substitute ( f(x) = \sqrt{x} ) into the definition.
Simplify the expression and compute the limit as ( h ) approaches 0.

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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.