Calculator Inputs
Formula Used
This calculator applies standard derivative rules to structured function families. It supports the power rule, chain rule, exponential differentiation, logarithmic differentiation, and basic trigonometric derivatives.
- Polynomial: d/dx(axn) = a·n·xn-1
- Power form: d/dx[a(bx+c)n] = a·n·b(bx+c)n-1
- Exponential: d/dx[a·ebx+c] = a·b·ebx+c
- General exponential: d/dx[a·kbx+c] = a·b·ln(k)·kbx+c
- Logarithmic: d/dx[a·ln(bx+c)] = a·b/(bx+c)
- Trigonometric: d/dx[a·sin(bx+c)] = a·b·cos(bx+c), and similar paired rules.
The second derivative is also computed to show curvature and concavity behavior near the selected evaluation point.
How to Use This Calculator
- Choose the function family that matches your expression structure.
- Enter the required coefficients, constants, and exponent or base values.
- Set the x-value where you want the derivative evaluated.
- Pick a sample table step size for nearby points.
- Submit the form to see the function, first derivative, second derivative, slope, tangent line, and normal line.
- Use the export buttons to download the result table as CSV or PDF.
Example Data Table
Example using f(x) = 3x² + 2x - 1 at x = 2.
| x | f(x) | f′(x) = 6x + 2 | f′′(x) = 6 |
|---|---|---|---|
| 1.0 | 4 | 8 | 6 |
| 1.5 | 8.75 | 11 | 6 |
| 2.0 | 15 | 14 | 6 |
| 2.5 | 22.75 | 17 | 6 |
| 3.0 | 32 | 20 | 6 |
FAQs
1. What functions can this calculator differentiate?
It supports structured polynomial, power, exponential, logarithmic, and trigonometric forms. Each family uses guided inputs instead of free-form symbolic text, which keeps outputs reliable and readable.
2. Does it return only the derivative expression?
No. It also gives the function value, first derivative value, second derivative value, tangent line, normal line, behavior near the point, and a nearby sample table.
3. Why is the second derivative useful?
The second derivative helps describe curvature. Positive values suggest upward concavity, while negative values suggest downward concavity near the selected x-value.
4. When can logarithmic results become undefined?
Logarithmic outputs are undefined when the inner expression bx + c is zero or negative. The calculator flags that restriction in the notes area.
5. Are trigonometric inputs treated as degrees?
No. The trigonometric options use radians. If your values are in degrees, convert them to radians before evaluating the function.
6. What happens when the tangent slope equals zero?
A zero slope means the tangent line is horizontal at that point. In that case, the normal line becomes vertical and is shown as x = constant.
7. Can I export the calculated results?
Yes. After submitting the form, you can download the nearby result table as CSV and save a formatted PDF summary for records or teaching material.
8. Does this tool handle unrestricted algebraic expressions?
No. It focuses on common structured families with precise parameter inputs. That design improves stability and keeps derivative formulas transparent.