# Using the limit definition, how do you find the derivative of # f ( x) = x^4#?

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Here is an alternative using one form of the limit definition.

I am using the limit definition in the form

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

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

Substitute (f(x) = x^4) into this formula:

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

Expand ((x + h)^4):

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

Substitute this expansion into the formula:

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

Cancel (x^4) terms:

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

Factor out (h) from the numerator:

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

Cancel (h) from the numerator and denominator:

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

Now, take the limit as (h) approaches 0:

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

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

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