Calculator Input
Plotly Graph
The chart compares the exponential function, its first derivative, and the tangent line near the selected evaluation point.
Example Data Table
| Case | Type | Expression | x | f(x) | f′(x) | Interpretation |
|---|---|---|---|---|---|---|
| 1 | Natural | 2e^(3x+1)+0 | 0 | 5.4366 | 16.3097 | Positive slope shows rapid growth at the origin. |
| 2 | General | 4·2^(0.5x-1)+3 | 2 | 7 | 1.3863 | Moderate growth because ln(2) scales the derivative. |
| 3 | Natural | -1.5e^(-2x+0.2)+1 | 1 | 0.7534 | 0.4932 | Negative coefficient and negative exponent still give a positive slope here. |
Formula Used
Natural exponential: For f(x) = a·e^(mx+n) + d, the derivative is f′(x) = a·m·e^(mx+n). The constant vertical shift d disappears during differentiation.
General base exponential: For f(x) = a·b^(mx+n) + d, the derivative is f′(x) = a·b^(mx+n)·ln(b)·m. This uses the rule for differentiating powers with a constant base.
Second derivative: The tool also computes curvature. For natural exponentials, f″(x) = a·m²·e^(mx+n). For general bases, f″(x) = a·b^(mx+n)·(ln(b)·m)².
Tangent line: At x = x₀, the tangent line is y = f′(x₀)(x - x₀) + f(x₀). This gives the best linear approximation near the chosen point.
How to Use This Calculator
- Select either the natural exponential form or the general base form.
- Enter the coefficient, inner linear values, and optional vertical shift.
- Provide the x-value where you want the derivative evaluated.
- Choose the graph range and the number of plotting points.
- Press Calculate Derivative to generate values, formulas, tangent line details, and the graph.
- Use the CSV and PDF buttons to export the result summary for reports or study notes.
FAQs
1. What does this calculator differentiate?
It differentiates exponential functions in two common forms: natural exponentials using e and general exponentials using any valid positive base except 1.
2. Why is the chain rule used here?
The exponent contains an inner expression such as mx + n. Differentiating the outer exponential requires multiplying by the derivative of that inner expression.
3. What happens to the vertical shift d?
A constant vertical shift changes the function value but not the derivative. Its derivative is always zero.
4. Why must the general base be positive and not 1?
The derivative formula uses ln(b). Logarithms require a positive base, and base 1 produces a constant function with no useful exponential growth behavior.
5. What does the second derivative mean?
The second derivative measures curvature. It helps show whether the function bends upward strongly and how quickly the slope itself changes.
6. What is elasticity in this output?
Elasticity estimates the proportional response of the function to x at the chosen point. It is useful when comparing local sensitivity across scales.
7. Why graph the derivative with the original function?
Seeing both lines together makes slope behavior easier to understand. It shows where the function grows quickly, slowly, or changes direction.
8. Can I use this for study notes or reports?
Yes. The calculator includes a readable summary table and lets you export the current results to CSV or PDF for classwork, revision, or documentation.