Using the limit definition, how do you find the derivative of #y = x^2 + x + 1 #?

Answer 1

#lim h ->0 ( [(a+h)^2+(a+h)+1]-[a^2+a+1))/h#
# lim h->0 (a^2+2ah+h^2+a+h+1-a^2-a-1)/h = lim h ->0 (2ah+h^2+h)/h-> lim h ->0 (h(2a+h+1))/h = 2a +1#

Find #f(a+h) = (a+h)^2+(a+h)+1 = a^2+2ah+h^2+a+h+1# and #f(a) =a^2+a+1 # then plug it in to the formula # lim h ->0 ( f(a+h)-f(a))/h # and then simplify. Remember to put in 0 for h after you factor out h from the numerator and cancel it to the h in the denominator. The remaining terms are your answer.
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Answer 2

To find the derivative of ( y = x^2 + x + 1 ) using the limit definition, follow these steps:

  1. Begin with the function ( y = x^2 + x + 1 ).
  2. Use the definition of the derivative, which states that ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ).
  3. Substitute the function ( f(x) = x^2 + x + 1 ) into the definition.
  4. Expand the function using algebra.
  5. Apply the limit as ( h ) approaches 0.
  6. Simplify the expression to find the derivative.

After simplification, the derivative of ( y = x^2 + x + 1 ) using the limit definition is ( y' = 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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