Using the limit definition, how do you find the derivative of #f(x) = 3x^2 + 8x + 4 #?

Answer 1

#f'(x)=6x+8#; see below for explanation

The limit definition of the derivative states that for a function #f(x)# its derivative equals
#f'(x)=lim_(hrarr0)(f(x+h)-f(x))/h#
So, when #f(x)=3x^2+8x+4#, we see that #f(x+h)=3(x+h)^2+8(x+h)+4#.

Applying the limit definition, we obtain

#f'(x)=lim_(hrarr0)(3(x+h)^2+8(x+h)+4-(3x^2+8x+4))/h#
Find #(x+h)^2#. Distribute the negative into #-(3x^2+4x+8)#.
#f'(x)=lim_(hrarr0)(3(x^2+2hx+h^2)+8(x+h)+4-3x^2-8x-4)/h#
Distribute the #3# and the #8#.
#f'(x)=lim_(hrarr0)(3x^2+6hx+3h^2+8x+8h+4-3x^2-8x-4)/h#

Cancel all like terms.

#f'(x)=lim_(hrarr0)(color(red)(cancel(color(black)(3x^2)))+6hx+3h^2color(blue)(cancel(color(black)(+8x)))+8hcolor(green)(cancel(color(black)(+4)))color(red)(cancel(color(black)(-3x^2)))color(blue)(cancel(color(black)(-8x)))color(green)(cancel(color(black)(-4))))/h#
#f'(x)=lim_(hrarr0)(6hx+3h^2+8h)/h#
Divide #h# from each term.
#f'(x)=lim_(hrarr0)6x+3h+8#
To evaluate the limit, plug in #0# for #h#.
#f'(x)=6x+3(0)+8#
#f'(x)=6x+8#
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Answer 2

To find the derivative of ( f(x) = 3x^2 + 8x + 4 ) using the limit definition, follow these steps:

  1. Begin with the definition of the derivative: ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ).
  2. Substitute the given function ( f(x) = 3x^2 + 8x + 4 ) into the formula.
  3. Expand the function ( f(x + h) ) and ( f(x) ).
  4. Simplify the expression by combining like terms.
  5. Take the limit as ( h ) approaches 0.
  6. Calculate the derivative.
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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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