AI Calculus Solver
Calculus asks how fast something changes. This AI calculus solver differentiates a function term by term, tells you which rule it used and simplifies the result, so you learn the technique instead of only collecting an answer.
How the calculus solver finds derivatives
Enter an expression after d/dx, for example d/dx 4x^3 − 6x. The built-in symbolic engine differentiates it exactly and then simplifies. The explanation names the rules involved so you can match them to your textbook.
| Rule | Formula | Use it when |
|---|---|---|
| Power rule | d/dx(xⁿ) = n·xⁿ⁻¹ | Any power of x |
| Constant multiple | d/dx(k·f) = k·f′ | A number multiplies the function |
| Sum rule | (f + g)′ = f′ + g′ | Terms are added or subtracted |
| Product rule | (fg)′ = f′g + fg′ | Two functions are multiplied |
| Quotient rule | (f/g)′ = (f′g − fg′) / g² | One function is divided by another |
| Chain rule | d/dx f(g(x)) = f′(g)·g′ | A function sits inside another |
Using derivatives: slope, maxima and minima
The derivative gives the slope of the curve at every point. Setting it to zero locates the turning points, and the sign of the second derivative tells you whether each is a peak or a valley. This is the standard method for optimisation questions, such as finding the largest area or the lowest cost.
To plot the function itself, switch to the graphing solver. For the reverse process, finding the original function from its rate of change, use the integral solver.
Who this calculus tool is for
It suits first-year university students, A-level and AP Calculus learners, and anyone refreshing their skills for engineering, economics or data science. If you are brand new to the topic, start with our guide on how to find a derivative and return here to check your own working.
Worked examples
Example 1: Differentiate f(x) = 4x³ − 6x
- Power rule on 4x³: 3 · 4x² = 12x².
- Power rule on −6x: −6.
Final answer: f′(x) = 12x² − 6
Example 2: Differentiate x²·sin x (product rule)
- Let f = x² and g = sin x, so f′ = 2x and g′ = cos x.
- Apply (fg)′ = f′g + fg′: 2x·sin x + x²·cos x.
Final answer: 2x sin x + x² cos x
Example 3: Differentiate (3x + 1)⁴ (chain rule)
- The outer function is u⁴, so its derivative is 4u³.
- The inner function is 3x + 1, with derivative 3.
- Multiply: 4(3x + 1)³ · 3.
Final answer: 12(3x + 1)³
Example 4: Find and classify the turning points of f(x) = x³ − 3x
- f′(x) = 3x² − 3. Set it to zero: x² = 1, so x = 1 or x = −1.
- f″(x) = 6x. At x = 1, f″ = 6 > 0, a minimum, with f(1) = −2.
- At x = −1, f″ = −6 < 0, a maximum, with f(−1) = 2.
Final answer: Minimum at (1, −2); maximum at (−1, 2)
Common mistakes to avoid
- Forgetting the chain rule’s “inner derivative”, so (3x + 1)⁴ becomes 4(3x + 1)³ instead of 12(3x + 1)³.
- Differentiating a product by multiplying the two derivatives together.
- Treating a constant such as −5 as if it had a derivative of −5; it is 0.
- Lowering the power but forgetting to multiply by the old power.
- Mixing up the order of the quotient rule: the numerator is f′g − fg′, never the other way round.
Frequently asked questions
What does d/dx mean?
It means “the derivative with respect to x”: how quickly the expression changes when x changes. The solver accepts forms like d/dx, derivative of or diff before your function.
Can the solver do second derivatives?
Yes. Differentiate once, then differentiate the result again, or ask the AI mode for the second derivative directly. It is used to test maxima and minima.
Does it handle trig, exponential and log functions?
Yes. Functions such as sin x, cos x, eˣ and ln x are supported in the derivative engine, and the AI mode covers more unusual forms.
Can I use it for integrals and limits?
Integrals have their own page, the integral solver. Limits and other advanced calculus questions can be sent through the AI mode when an API key is configured.
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AI-generated solutions are for learning purposes. Always verify important answers.