How do you use the limit definition of the derivative to find the derivative of #f(x)=sqrt(x)-2x+1#?

Answer 1

#f'(x)=1/(2sqrt(x))-2#

The limit definition says that for #f(x)#, #f'(x)=lim_(h->0)(f(x+h)-f(x))/h#
For #f(x)=sqrt(x)-2x+1#, we have: #f'(x)=lim_(h->0)(sqrt(x+h)-2(x+h)+1-sqrt(x)+2x-1)/h#
#color(white)(llllllll)=lim_(h->0)(sqrt(x+h)-2x-2h+1-sqrt(x)+2x-1)/h#
#color(white)(llllllll)=lim_(h->0)(sqrt(x+h)-2h-sqrt(x))/h#
#color(white)(llllllll)=lim_(h->0)(sqrt(x+h)-sqrt(x))/h-lim_(h->0)(2h)/h#
#color(white)(llllllll)=lim_(h->0)1/(h/(sqrt(x+h)-sqrt(x)))-2#
#color(white)(llllllll)=lim_(h->0)1/((h(sqrt(x+h)+sqrt(x)))/((sqrt(x+h)-sqrt(x))(sqrt(x+h)+sqrt(x))))-2#
#color(white)(llllllll)=lim_(h->0)((sqrt(x+h)-sqrt(x))(sqrt(x+h)+sqrt(x)))/(h(sqrt(x+h)+sqrt(x)))-2#
#color(white)(llllllll)=lim_(h->0)(x+h-x+sqrt(x^2+hx)-sqrt(x^2+hx))/(hsqrt(x+h)+hsqrt(x))-2#
#color(white)(llllllll)=lim_(h->0)h/(hsqrt(x+h)+hsqrt(x))-2#
#color(white)(llllllll)=lim_(h->0)1/(sqrt(x+h)+sqrt(x))-2#
#color(white)(llllllll)=1/(2sqrt(x))-2#
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Answer 2

To find the derivative of ( f(x) = \sqrt{x} - 2x + 1 ) using the limit definition of the derivative, follow these steps:

  1. Start with the function ( f(x) ).
  2. Write down the limit definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ).
  3. Substitute the function ( f(x) = \sqrt{x} - 2x + 1 ) into the limit definition.
  4. Simplify the expression.
  5. Take the limit as ( h ) approaches 0.
  6. Calculate the derivative ( f'(x) ).

This process will yield the derivative of the function ( f(x) = \sqrt{x} - 2x + 1 ).

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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.

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.

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.

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.

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