# How do you use the definition of a derivative to find the derivative of #f ( x) = x^4#?

I found

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To find the derivative of ( f(x) = x^4 ) using the definition of a derivative, we apply the definition:

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

Substitute ( f(x) = x^4 ) into the definition:

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

Expand ( (x + h)^4 ) using the binomial theorem:

[ (x + h)^4 = x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4 ]

Substitute this expression into the derivative definition:

[ f'(x) = \lim_{h \to 0} \frac{(x^4 + 4x^3h + 6x^2h^2 + 4xh^3 + h^4) - x^4}{h} ]

[ f'(x) = \lim_{h \to 0} \frac{4x^3h + 6x^2h^2 + 4xh^3 + h^4}{h} ]

[ f'(x) = \lim_{h \to 0} 4x^3 + 6x^2h + 4xh^2 + h^3 ]

[ f'(x) = 4x^3 ]

So, the derivative of ( f(x) = x^4 ) is ( f'(x) = 4x^3 ).

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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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