Enter f(x) and Point(s) x0
d/dx |f(x)| at x = x0Results
Steps and Piecewise Rule
d/dx |f(x)| = ( f(x) / |f(x)| ) · f'(x), for f(x) ≠ 0
Not differentiable at x where f(x) = 0.
Formula Used
Let y = |f(x)| where f is differentiable near x0. If f(x0) ≠ 0 then y'(x0) = sign(f(x0)) × f'(x0). If f(x0) = 0, the function is not differentiable there.
- sign(u) = u/|u| for u ≠ 0.
- Polynomials: use the power rule to obtain f'(x).
- Zeroes of f are corners of |f(x)| and break differentiability.
Neighborhood Scan and Chart
We search for sign changes of f around x0. Potential non‑differentiable points of |f| are highlighted.
Example Data
| f(x) inner | x0 | f(x0) | f'(x0) | y'(x0) = d/dx |f(x)| | One‑sided (L, R) | Note |
|---|---|---|---|---|---|---|
| x | 1 | 1 | 1 | 1 | (1, 1) | sign(+)×1 = 1 |
| x | -2 | -2 | 1 | -1 | (-1, -1) | sign(-)×1 = -1 |
| x-3 | 3 | 0 | 1 | undefined | (-1, 1) | Corner at zero of inner |
| 2x+1 | -1 | -1 | 2 | -2 | (-2, -2) | sign(-)×2 = -2 |
| x^2-4 | 1 | -3 | 2 | -2 | (-2, -2) | sign(-)×2 = -2 |
Rows illustrate one‑sided behavior and non‑differentiability at zeros.
Reference & Insights
1) Key Concepts
- Derivative rule: d/dx |f(x)| = sign(f(x))·f′(x) if f(x) ≠ 0.
- Non‑differentiable at zeros of f(x); check one‑sided limits.
- For linear f(x)=mx+b, derivative magnitude equals |m| piecewise.
- Continuity holds everywhere; differentiability fails where f(x)=0.
2) Piecewise Behavior & Differentiability
| Inner f(x) | Condition | d/dx |f(x)| | Differentiable? |
|---|---|---|---|
| x | x > 0 | +1 | Yes |
| x | x < 0 | −1 | Yes |
| x | x = 0 | undefined | No |
| x−a | x > a | +1 | Yes |
| x−a | x < a | −1 | Yes |
| x−a | x = a | undefined | No |
3) Worked Mini‑Examples
- y=|2x−3| at x=5 → f=7, f′=2 ⇒ y′=+2.
- y=|2x−3| at x=1.5 → f=0 ⇒ not differentiable.
- y=|x^2−4| at x=3 → f=5, f′=6 ⇒ y′=+6.
- y=|x^2−4| at x=−2 → f=0 ⇒ not differentiable.
4) Practical Tips
- Use the neighborhood scan to detect zeros of f quickly.
- Enable one‑sided derivatives to verify a corner at zeros.
- Increase scan resolution for highly oscillatory polynomials.
- Numeric expression mode uses a five‑point stencil for f′(x).
FAQs
1) When is d/dx |f(x)| undefined?
It is undefined where the inner function f(x) equals zero. Those points are corners of |f(x)|. Check one‑sided derivatives; if they differ, the derivative does not exist there.
2) How do you handle general expressions?
Enter f(x) as an expression and the tool uses numerical differentiation for f′(x), then applies the sign rule. This works well near regular points away from zeros of f(x).
3) What if results look noisy?
Increase decimal places modestly and raise scan resolution. For expressions with rapid oscillations or steep slopes, smaller neighborhoods and careful x₀ choices often stabilize the derivative estimate.
4) Why show one‑sided derivatives?
At zeros of f(x), |f(x)| often forms a corner. One‑sided derivatives reveal the left and right limits. If they disagree, the derivative at that point does not exist.
5) Can this detect all zeros near x₀?
The neighborhood scan reports sign changes and refines candidates by bisection. Very tight or repeated roots may require increasing resolution or adjusting the scan interval.
6) Is continuity affected at zeros?
|f(x)| remains continuous even when f(x) crosses zero, but differentiability fails there. The graph touches with a sharp corner, which the tool highlights through scans and one‑sided values.
Short Glossary
- Sign
- The sign of a number is +1 for positive values and −1 for negative values. For derivatives of |f(x)|, use sign(f(x))·f′(x) when f(x) ≠ 0.
- Corner
- A sharp point on the graph where left and right derivatives differ. For |f(x)|, corners occur at zeros of the inner function f(x).
- Piecewise
- A function defined by different formulas on different intervals. |f(x)| behaves piecewise: multiply f′(x) by +1 or −1 depending on the sign of f(x).