Calculator input
Example data table
| Example expression | Derivative | Main rule |
|---|---|---|
| x^3 + 2*x | 3*x^2 + 2 | Power rule |
| sin(x)*x^2 | cos(x)*x^2 + sin(x)*2*x | Product rule |
| log(x) | 1/x | Logarithmic rule |
| exp(x^2) | exp(x^2)*2*x | Chain rule |
Formula used
This calculator applies symbolic differentiation rules to the parsed expression tree. It identifies operators, functions, nested terms, and variable dependencies before generating a derivative expression.
Core rules used:
- Power rule: d/dx[xn] = n·xn-1
- Sum rule: d/dx[u + v] = u′ + v′
- Product rule: d/dx[u·v] = u′v + uv′
- Quotient rule: d/dx[u/v] = (u′v - uv′)/v2
- Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x)
- Special functions: sin, cos, tan, exp, ln, log, sqrt, and inverse trig functions are differentiated with their standard formulas.
After differentiation, the result is simplified by removing zeros, ones, repeated negatives, and easy constant operations where possible.
How to use this calculator
- Enter your mathematical expression using supported operators and functions.
- Choose the variable for differentiation, usually x.
- Set the graph range and point count for plotting.
- Press Differentiate expression to compute the symbolic derivative.
- Review the readable derivative, rules used, graph, and generated sample values.
- Use the CSV or PDF buttons to export the result summary.
FAQs
1. What expressions can this calculator handle?
It supports addition, subtraction, multiplication, division, exponents, parentheses, common constants, and many standard functions such as trigonometric, logarithmic, exponential, and radical forms.
2. Does it return symbolic results instead of numeric slopes?
Yes. The calculator builds an expression tree and returns a symbolic derivative. It also plots the original function and derivative numerically for visualization.
3. Can I differentiate nested functions?
Yes. Nested inputs like exp(x^2), sin(log(x)), and sqrt(x^3 + 1) are handled using the chain rule when the inside term depends on the chosen variable.
4. Why do some graph points disappear?
Points are skipped when the function or derivative is undefined, such as log of negative values, square roots of negatives, or division by zero.
5. Does the simplifier always produce the shortest answer?
Not always. It performs practical simplifications like removing zero terms and constant combinations, but it is not a full computer algebra system for every identity.
6. Can I use variables other than x?
Yes. Enter a variable such as t or y. Any different variable names inside the expression are treated as constants during differentiation.
7. What is included in the CSV and PDF export?
The export includes the input expression, derivative, selected variable, graph domain, and the generated sample value table for easy reporting or review.
8. Is this useful for learning calculus?
Yes. It highlights which derivative rules were used, shows the resulting form, and lets you compare the original function against its slope behavior visually.