# For what values of x is the function #f(x)=abs(x^2-9)# differentiable?

This transition occurs when:

graph{|x^2-9| [-20.04, 19.96, -3.16, 16.84]}

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The function f(x) = |x^2 - 9| is differentiable for all values of x except where the derivative does not exist. Since absolute value functions are continuous everywhere, the function is differentiable everywhere except at the points where the derivative is undefined.

To find where the function is not differentiable, we need to examine where the derivative has discontinuities. The derivative of f(x) can be found using the chain rule and the derivative of the absolute value function.

The derivative of f(x) is:

f'(x) = (x^2 - 9) / |x^2 - 9| * 2x.

The function is not differentiable when the denominator of this expression is equal to zero, which occurs when x^2 - 9 = 0. Solving this equation, we find that x = ±3.

So, the function f(x) = |x^2 - 9| is differentiable for all values of x except x = ±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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