How do you find the f'(x) using the formal definition of a derivative if #f(x)= 2x^2  3x+4#?
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To find ( f'(x) ) using the formal definition of a derivative if ( f(x) = 2x^2  3x + 4 ):

Recall the formal definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ).

Substitute the function ( f(x) = 2x^2  3x + 4 ) into the definition: ( f'(x) = \lim_{h \to 0} \frac{(2(x+h)^2  3(x+h) + 4)  (2x^2  3x + 4)}{h} ).

Expand and simplify the expression: ( f'(x) = \lim_{h \to 0} \frac{2(x^2 + 2hx + h^2)  3x  3h + 4  2x^2 + 3x  4}{h} ).

Combine like terms: ( f'(x) = \lim_{h \to 0} \frac{2x^2 + 4hx + 2h^2  3x  3h + 4  2x^2 + 3x  4}{h} ). ( f'(x) = \lim_{h \to 0} \frac{4hx + 2h^2  3h}{h} ).

Cancel common terms: ( f'(x) = \lim_{h \to 0} 4x + 2h  3 ).

Evaluate the limit as ( h ) approaches 0: ( f'(x) = 4x  3 ).
Therefore, the derivative of ( f(x) = 2x^2  3x + 4 ) with respect to ( x ) is ( f'(x) = 4x  3 ).
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