Inputs
Tip: The derivative exists for g(x) > 0 only. At g(x)=0 it's undefined; for g(x)<0, √g(x) is non-real.
Results
Chain rule: g′(x) / [2√g(x)]
At x = —
g(x): —
√g(x): —
g′(x) numeric: —
d/dx √g(x): —
Expression
g(x) = —
Notes
Provide inputs and press Calculate.
| # | x | g(x) | √g(x) | g′(x) ≈ | d/dx √g(x) | Note |
|---|
Numerical derivative uses central difference: [g(x+h) − g(x−h)] / (2h).
Symbolic Derivatives (Common Forms)
Using rule-based differentiation with simplification:
g′(x) = —
d/dx √g(x) = —
If unsupported features appear, a simplified symbolic form may be unavailable.
Plots
Interactive
Lines show g(x), √g(x), g′(x) (numeric), and d/dx √g(x) (where defined).
Formula Used
For a differentiable inner function g(x) with g(x) > 0,
d/dx √g(x) = g′(x) / (2 √g(x))
We estimate g′(x) numerically by central difference:
g′(x) ≈ [ g(x + h) − g(x − h) ] / (2h)
Choose a small h. This tool adapts h to x if left blank.
How to Use
- Enter the inner function g(x) using standard math notation.
- Optionally set a single point x for evaluation.
- To generate a table, set start, end, and step.
- Leave h empty for adaptive accuracy, or set a custom value.
- Click Calculate to see values, symbolic forms, and plots.
- Export your table with Download CSV or Download PDF.
- Use Copy Shareable URL to save or share your setup.
Example Data Table g(x)=x^2+4x+4
| x | g(x) | √g(x) | g′(x) | d/dx √g(x) | Note |
|---|---|---|---|---|---|
| -3 | 1 | 1 | -2 | -1 | Sign since √((x+2)^2) = |x+2| |
| -2 | 0 | 0 | 0 | undefined | Derivative undefined at kink g(x)=0 |
| -1 | 1 | 1 | 2 | 1 | Right of kink, slope becomes +1 |
| 0 | 4 | 2 | 4 | 1 | g(x)>0 so derivative well-defined |
| 1 | 9 | 3 | 6 | 1 | Consistent with chain rule |
This example shows the domain and a non-differentiable point at x = −2.
Reference: Common g(x) & Closed-Form d/dx √g(x)
| g(x) | g′(x) | d/dx √g(x) | Domain (reals) |
|---|---|---|---|
| x | 1 | 1 / (2√x) | x > 0 |
| x^2 + 1 | 2x | x / √(x^2 + 1) | All x |
| e^x | e^x | √(e^x) / 2 | All x |
| ln(x) + 1 | 1 / x | 1 / [ 2x√(ln(x) + 1) ] | x > e^{-1} |
| sin(x) + 2 | cos(x) | cos(x) / [ 2√(sin(x) + 2) ] | All x |
Use these for quick checks against numeric outputs and plots.
Domain & Differentiability Guide
| Condition on g(x) | Interpretation | Derivative status | Example at x = a |
|---|---|---|---|
| g(x) > 0 | √g(x) is real and smooth | Defined: g′(a)/(2√g(a)) | g(x)=x^2+1, a=0 ⇒ 0/√1 = 0 |
| g(x) = 0 | Corner/cusp in √g(x) | Undefined in reals | g(x)=(x+2)^2, a=-2 |
| g(x) < 0 | √g(x) is non-real | Not defined over reals | g(x)=x-1, a=0 ⇒ negative |
| g not differentiable at a | Chain rule fails at a | Undefined | g(x)=|x|, a=0 |
Ensure both g(x) and √g(x) meet real-domain requirements before interpreting results.
FAQs
Use numbers, x, parentheses, and functions: sqrt, sin, cos, tan, log, exp, abs, min, max, etc. Operators: +, -, *, /, ^. Constants like pi and e are supported.
The derivative of √g(x) needs g(x)>0. If g(x)=0, √g(x) has a cusp or corner and the derivative does not exist. If g(x)<0, the square root is non-real in the reals.
Central differences are second-order accurate. We adapt h to scale with x unless you specify it. Extremely small h may amplify rounding errors; very large h reduces accuracy.
Yes. Leave the range empty and fill just x; press Calculate. The tool reports g(x), √g(x), g′(x), and d/dx √g(x) at that point.
Trigonometric functions use radians. If you need degrees, convert inputs using x * pi/180 inside your expression.
After calculating, use the green button for CSV. Use the red button to generate a formatted PDF with your inputs and the full table.
This tool focuses on real analysis. For g(x)<0, √g(x) is complex; we mark those rows and skip the derivative in the real numbers.