Question

find the one-sided derivatives of the function \\(f(x) = |x + 25|\\) at the point \\(x = -25\\), if they exist. if the derivative does not exist, write dne for your answer.

left-hand derivative at \\(x = -25\\):
right-hand derivative at \\(x = -25\\):

Explanation:

Define the piecewise function

$$ f(x) = |x + 25| =
LATEXBLOCK0
$$

Calculate the left-hand derivative

$$ f'_{-}(-25) = \lim_{h \to 0^-} \frac{f(-25 + h) - f(-25)}{h} = \lim_{h \to 0^-} \frac{-h - 0}{h} = -1 $$

Calculate the right-hand derivative

$$ f'_{+}(-25) = \lim_{h \to 0^+} \frac{f(-25 + h) - f(-25)}{h} = \lim_{h \to 0^+} \frac{h - 0}{h} = 1 $$

Answer:

Find the one-sided derivatives of the function \(f(x) = |x + 25|\) at the point \(x = -25\), if they exist. If the derivative does not exist, write DNE for your answer.

Left-hand derivative at \(x = -25\): <blank>-1</blank>

Right-hand derivative at \(x = -25\): <blank>1</blank>