How do you find f'(x) using the limit definition given #f(x) = 3 x^3 + 9 x + 4#?
Use
Write the simplest form of f(x+h)
Subtract f(x) from that.
A common factor of
Let
Use that form to simplify the numerator:
Remove a common factor, h:
Substitute the simplified numerator into the limit:
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To find ( f'(x) ) using the limit definition for the function ( f(x) = 3x^3 + 9x + 4 ), follow these steps:

Write down the limit definition of the derivative: [ f'(x) = \lim_{h \to 0} \frac{f(x + h)  f(x)}{h} ]

Substitute the given function ( f(x) = 3x^3 + 9x + 4 ) into the limit definition.

Expand and simplify the expression.

Take the limit as ( h ) approaches 0.

Simplify the expression further if possible to obtain ( f'(x) ).
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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