# How do you find f'(x) using the definition of a derivative for #f(x)=5.5x^2 - x + 4.2 #?

See the explanation.

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To find ( f'(x) ) using the definition of a derivative for ( f(x) = 5.5x^2 - x + 4.2 ), we use the formula:

[ f'(x) = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} ]

Substitute ( f(x) = 5.5x^2 - x + 4.2 ) into the formula and simplify:

[ f'(x) = \lim_{{h \to 0}} \frac{{5.5(x + h)^2 - (x + h) + 4.2 - (5.5x^2 - x + 4.2)}}{h} ]

[ = \lim_{{h \to 0}} \frac{{5.5(x^2 + 2xh + h^2) - x - h + 4.2 - 5.5x^2 + x - 4.2}}{h} ]

[ = \lim_{{h \to 0}} \frac{{5.5x^2 + 11xh + 5.5h^2 - x - h + 4.2 - 5.5x^2 + x - 4.2}}{h} ]

[ = \lim_{{h \to 0}} \frac{{11xh + 5.5h^2 - h}}{h} ]

[ = \lim_{{h \to 0}} \frac{{h(11x + 5.5h - 1)}}{h} ]

[ = \lim_{{h \to 0}} (11x + 5.5h - 1) ]

[ = 11x - 1 ]

So, ( f'(x) = 11x - 1 ).

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To find ( f'(x) ) using the definition of a derivative for ( f(x) = 5.5x^2 - x + 4.2 ), we use the definition of the derivative:

[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} ]

Substitute the function ( f(x) ) into this expression:

[ f'(x) = \lim_{h \to 0} \frac{5.5(x+h)^2 - (x+h) + 4.2 - (5.5x^2 - x + 4.2)}{h} ]

Expand and simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{5.5(x^2 + 2xh + h^2) - x - h + 4.2 - 5.5x^2 + x - 4.2}{h} ] [ f'(x) = \lim_{h \to 0} \frac{5.5x^2 + 11xh + 5.5h^2 - x - h + 4.2 - 5.5x^2 + x - 4.2}{h} ] [ f'(x) = \lim_{h \to 0} \frac{11xh + 5.5h^2 - h}{h} ] [ f'(x) = \lim_{h \to 0} \frac{h(11x + 5.5h - 1)}{h} ] [ f'(x) = \lim_{h \to 0} (11x + 5.5h - 1) ]

Now, evaluate the limit as ( h \to 0 ):

[ f'(x) = 11x - 1 ]

Therefore, the derivative of ( f(x) ) with respect to ( x ), denoted as ( f'(x) ), is ( 11x - 1 ).

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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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