Limits, Continuity & Differentiability
Domain of Derivatives
Grade 12
Question:
<p>The domain of the derivative of the functions <i>f(x) =</i></p><p>$$f(x) = \begin{cases} \tan^{-1} x, & \text{if } |x| > 1 \\ \frac{1}{2}(|x| - 1), & \text{if } |x| \leq 1 \end{cases}$$</p><p>is</p>
<p>(a) <i>ℝ \ {0}</i></p>
<p>(b) <i>ℝ \ {1}</i></p>
<p>(c) <i>ℝ \ {−1}</i></p>
<p>(d) <i>ℝ \ {−1, 1}</i></p>
Step-by-Step Solution
Key Concept: Check left and right derivatives at boundary points where function definition changes; domain of derivative excludes non-differentiable points
<p>Check differentiability at transition points <i>x = ±1</i>:</p><p>At <i>x = 1</i>:</p><p>Left derivative: <i>f'(1<sup>−</sup>) = 1/2</i></p><p>Right derivative: <i>f'(1<sup>+</sup>) = 1/(1+1²) = 1/2</i></p><p>Similarly at <i>x = −1</i>, the derivatives do not match (one is 1/2, other is −1/2)</p><p>Therefore derivative exists for all <i>x ∈ ℝ \ {−1, 1}</i></p>
Correct Answer: d