# Using the limit definition, how do you differentiate #f(x)=1-x^2#?

If

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To differentiate the function (f(x) = 1 - x^2) using the limit definition of a derivative, we apply the following formula:

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

Substitute the function (f(x) = 1 - x^2) into the formula:

[f'(x) = \lim_{h \to 0} \frac{(1 - (x + h)^2) - (1 - x^2)}{h}]

Expand and simplify:

[f'(x) = \lim_{h \to 0} \frac{1 - (x^2 + 2xh + h^2) - 1 + x^2}{h}]

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

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

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

Now, as (h) approaches 0, the limit simplifies to:

[f'(x) = -2x]

Therefore, the derivative of (f(x) = 1 - x^2) with respect to (x) is (f'(x) = -2x).

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