Describe the x-values at which the function is differentiable. (Enter your answer using interval notation.) y = |x^2 − 81|

Question

^ 2 exponent 2

Answer 1

Find dy/dx

y = |x²-81|

dy/dx = 2x•(x²-81)/|x²-81|

dy/dx = 2x•(x²-81)/|x²-81|

Since |x²-81| = |(x+9)(x-9)| ≠ 0, then f is not differentiable at x=9 and x=-9. This makes the x-values at which f is differentiable (-∞,-9)∪(-9,9)∪(9,∞). Again, if you graph the derivative, you can see this for yourself because of the jump discontinuities at those points.

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