Derivative Calculator
Taking the derivative of x^3 sin(x) - ln(x)/x by hand is fine once; doing it forty times for a problem set is soul-crushing. Paste any algebraic expression, pick the variable, and this calculator returns the simplified derivative along with the product, quotient and chain rule steps so you can audit the working.
How to differentiate an expression
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1
Enter the expression
Use standard notation: `x^2 + 3*x`, `sin(x)/x`, `e^(2x)`, `ln(x)`. Implicit multiplication is accepted.
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2
Choose the variable
Differentiate with respect to x, t, theta or any letter; other letters are treated as constants.
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3
Pick the order
First, second or higher derivatives are supported; results chain automatically.
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4
Review steps
Expand the working to see which rule (power, product, quotient, chain) was applied at each step.
The rules the calculator applies
| Rule | Form | Example |
|---|---|---|
| Power | d/dx [x^n] = n * x^(n-1) | x^5 becomes 5x^4 |
| Constant | d/dx [c] = 0 | 7 becomes 0 |
| Sum | d/dx [f + g] = f’ + g’ | x^2 + x becomes 2x + 1 |
| Product | d/dx [f * g] = f’ g + f g’ | x sin(x) becomes sin(x) + x cos(x) |
| Quotient | d/dx [f / g] = (f’ g - f g’) / g^2 | sin(x)/x has a classic quotient output |
| Chain | d/dx [f(g(x))] = f’(g(x)) * g’(x) | sin(x^2) becomes 2x cos(x^2) |
| Exponential | d/dx [e^x] = e^x; d/dx [a^x] = a^x ln(a) | e^(2x) becomes 2 e^(2x) |
| Log | d/dx [ln(x)] = 1/x; d/dx [log_a(x)] = 1/(x ln a) | ln(3x) becomes 1/x |
| Trig | sin’ = cos, cos’ = -sin, tan’ = sec^2 | tan(x) becomes sec^2(x) |
Worked example
For f(x) = x^2 * ln(x):
- Identify product: u = x^2, v = ln(x).
- Apply product rule: f’ = u’v + uv’ = (2x)(ln x) + (x^2)(1/x).
- Simplify: f’ = 2x ln(x) + x.
The tool shows exactly these three lines in the step output, so you can copy them into your homework or proof.
Tips for clean input
- Use explicit multiplication in ambiguous cases: write
2*x*yrather than2xyto avoid parser confusion with function names. - Wrap exponents when they are more than a single character:
x^(2n+1), notx^2n+1. - Prefer
e^xoverexp(x)unless you need scientific notation inside the exponent. - Check for hidden constants.
f(x) = a*x^2 + bdifferentiates to2ax, not2ax + b', because the tool assumes every letter except your chosen variable is a constant.
Limitations
- Implicit differentiation (solving for dy/dx when y is implicit) is not handled; rearrange to an explicit form first.
- Piecewise functions need the sections entered separately.
- Very long expressions may take a fraction of a second to simplify the final form.
Frequently Asked Questions
Yes. If your expression has multiple variables, pick the one you want to differentiate with respect to; every other variable is held constant automatically, which is exactly what partial differentiation means.
The tool applies algebraic simplification rules such as combining like terms and canceling factors, but will not invent identities. If two forms are equivalent but non-obvious, both may appear as valid outputs.
Yes, up to the tenth order in the current build. Set the order and the calculator chains the first derivative rule repeatedly.
No. The computer-algebra engine runs in the browser (WebAssembly build). The expression you type stays on your device.
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