# How do you find the derivative of #f(x)=4x^2# using the limit definition?

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To find the derivative of ( f(x) = 4x^2 ) using the limit definition, we use the formula:

[ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} ]

Substitute ( f(x) = 4x^2 ) into the formula:

[ f'(x) = \lim_{h \to 0} \frac{(4(x + h)^2 - 4x^2)}{h} ]

Expand and simplify the expression:

[ f'(x) = \lim_{h \to 0} \frac{(4x^2 + 8xh + 4h^2 - 4x^2)}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{8xh + 4h^2}{h} ]

[ f'(x) = \lim_{h \to 0} (8x + 4h) ]

[ f'(x) = 8x ]

Therefore, the derivative of ( f(x) = 4x^2 ) with respect to ( x ) is ( f'(x) = 8x ).

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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.

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