Consider the following function.f(x) = |x − 9|Find the derivative from the left at x = 9. If it does not exist, enter NONE.Find the derivative from the right at x = 9. If it doe…

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asked Jan 17, 2023 141k views

1 Answer

Yachi · Answer 1 · 2023-01-23T18:52:24+0000

The form of the derivative of the absolute equation is

$\begin{gathered} f(x)=\lvert x-a\rvert \\ (dx)/(dy)=(\lvert x-a\rvert)/(x-a) \\ (dy)/(dx)=(x-a)/(\lvert x-a\rvert) \end{gathered}$

For the given function

$f(x)=\lvert x-9\rvert$

We will find the derivative from the left at x = 9

$(dy)/(dx)=(x-9)/(\lvert x-9\rvert)$

Substitute x by 9

$\begin{gathered} (dy)/(dx)=(9-9)/(\lvert9-9\rvert) \\ (dy)/(dx)=(0)/(0) \end{gathered}$

Then dy/dx does not exist (None)

The derivative from right at x = 9

$(dx)/(dy)=(\lvert x-9\rvert)/(x-9)$

Substitute x by 9

$\begin{gathered} (dx)/(dy)=(\lvert9-9\rvert)/(9-9) \\ (dx)/(dy)=(0)/(0) \end{gathered}$

Then dx/dy does not exist (None)