How do you use the definition of a derivative to find the derivative of #f(x)= 2x^2 - 3x+4#?

Question

How do you use the definition of a derivative to find the derivative of #f(x)=  2x^2 - 3x+4#?

Samantha Adkison · Accepted Answer

# f'(x) = 4x-3# By definition # f'(x) =lim_(hrarr0)( (f(x+h)-f(x))/h ) # So, with # f(x)=2x^2-3x+4 # we have: # f'(x) = lim_(hrarr0)( ( (2(x+h)^2-3(x+h)+4) - (2x^2-3x+4) ) / h ) # # :. f'(x) = lim_(hrarr0)( ( ( 2(x^2+2hx+h^2)-3x-3h+4) - (2x^2-3x+4) ) /h )# # :. f'(x) = lim_(hrarr0)( ( 2x^2+4hx+2h^2-3x-3h+4 - 2x^2+3x-4) /h )# # :. f'(x) = lim_(hrarr0)( ( 4hx+2h^2-3h) /h )# # :. f'(x) = lim_(hrarr0)( 4x+2h-3 )# # :. f'(x) = 4x-3#

Aurora Alvarado · Answer

undefined To find the derivative of $ f(x) = 2x^2 - 3x + 4 $, you use the definition of a derivative, which states that the derivative of a function $ f(x) $ at a point $ x $ is the limit of the difference quotient as the interval around $ x $ shrinks to zero. Applying this definition to $ f(x) $, you calculate the limit of the expression $ \frac{{f(x + h) - f(x)}}{h} $ as $ h $ approaches zero. By evaluating this limit, you obtain the derivative of $ f(x) $ with respect to $ x $. Applying this process to $ f(x) = 2x^2 - 3x + 4 $, you would first compute $ f(x + h) $, then find the difference quotient, and finally evaluate the limit as $ h $ tends to zero. After simplifying the expression and evaluating the limit, you will find the derivative of $ f(x) $, denoted as $ f'(x) $, which represents the rate of change of $ f(x) $ with respect to $ x $.