Using the limit definition, how do you find the derivative of #y = x^2 + x + 1 #?
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To find the derivative of ( y = x^2 + x + 1 ) using the limit definition, follow these steps:
- Begin with the function ( y = x^2 + x + 1 ).
- Use the definition of the derivative, which states that ( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ).
- Substitute the function ( f(x) = x^2 + x + 1 ) into the definition.
- Expand the function using algebra.
- Apply the limit as ( h ) approaches 0.
- 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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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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