Calculus Solver

Type a function, click Calculate. This calculus solver differentiates any function instantly, right on the page.

• Function — the box comes pre-filled with x^2 - 3x. Replace it with any function, or tap an example button such as sin(x), x^3, e^x, or 1/x
• With respect to — pick the variable: x, y, or t

Click Calculate and the calculus solver returns the derivative: for the pre-filled $x^2 - 3x$ it gives $2x - 3$ — how fast the function changes, at every point.

The form covers derivatives. The AI box below the form covers the rest of calculus: limits, integrals, related-rates and optimization word problems, full exam questions. It takes plain questions, photos of the problem, and follow-ups:

• Ask in plain words. “Evaluate the limit as x approaches 2 of (x² − 4)/(x − 2).” The AI solves it with every step shown.
• Take a photo of your homework. No typing: the AI reads the problem — limit, derivative, or integral — and solves it with full steps.
• Ask about the ideas, not just the answer. “Why do derivatives and integrals undo each other?” The AI explains — and any derivative it produces matches the calculus solver form above, so nothing is guessed.
• Keep asking follow-ups. “Now integrate my result” or “what does this mean graphically?” — this calculus solver page keeps teaching.

Under the hood, this calculus solver is two engines side by side.

The form is exact symbolic math. When you click Calculate, it parses the function you typed and applies the differentiation rules — power, sum, product, chain — symbolically. The result $2x - 3$ for $x^2 - 3x$ is not a rounded estimate; it is the exact derivative, the same one a textbook would derive line by line.

The AI box reads whole problems. A limit, a definite integral, a related-rates word problem — none of these fit one small form. The AI takes the full statement in plain words or a photo, decides which tool the problem needs, and solves it with the steps written out.

The two engines check each other. Any derivative the AI produces can be confirmed in the form above — type it and compare. That division of labor is the design of this calculus solver page: exact computation where a form suffices, explained reasoning where a form cannot.

Strip calculus to its frame and three ideas hold everything up.

Limits — the idea of approach. Some values cannot be found by plugging in: divide by zero and algebra stops. A limit asks what value a function closes in on as the input closes in on a target. Limits are the foundation that makes “instantaneous” and “infinitely many slices” mathematically precise — every 0/0 you meet is a limit in disguise.

Derivatives — the idea of change. The derivative measures how fast a quantity changes at a single instant: speed from position, slope of a curve at a point. It is what the form at the top of this page computes. Where it shows up: velocity and acceleration, growth rates of populations and interest, and optimization — set a derivative to zero to find the peaks and valleys, the cheapest box, the maximum profit.

Integrals — the idea of accumulation. The integral adds up infinitely many tiny contributions: slices of area under a curve, small steps of speed into total distance. Where it shows up: any total built from a rate — rainfall from a rain rate, energy from power, distance from a speedometer.

Learn the three separately, then watch them connect — the next block shows the deepest link in the whole subject.

The two heavy pillars undo each other. Integrate a derivative and you recover the original function:

$\int f'(x)\,dx=f(x)+C$

The $+ C$ is the one souvenir the round trip keeps — differentiation erases constants, so the return trip cannot restore them.

Watch the round trip with the numbers already in the form:

1. Differentiate (click Calculate above): $\dfrac{d}{dx}(x^2-3x)=2x-3$ — the power rule turns $x^2$ into $2x$, and $-3x$ into $-3$.
2. Integrate the result: $\int(2x-3)\,dx=x^2-3x+C$ — term by term, $2x$ back to $x^2$, $-3$ back to $-3x$.
3. You are home: the original function, plus the unavoidable constant.

This is the Fundamental Theorem of Calculus in its shortest form: differentiation and integration are reverse operations. Every calculus solver trick — recovering position from velocity, rebuilding a function from its slope — runs on this one line.

One limit shows everything the idea can do:

$\lim_{x\to2}\dfrac{x^2-4}{x-2}=4$

Try to substitute $x = 2$ directly: the top becomes $4 - 4 = 0$ and the bottom becomes $0$ — the fraction $\dfrac{0}{0}$ is undefined. Algebra is stuck. The limit asks a subtler question instead: what do the numerator and the denominator approach as $x$ closes in on 2 from both sides?

1. Factor the top: $x^2 - 4 = (x-2)(x+2)$.
2. Cancel: $\dfrac{(x-2)(x+2)}{x-2} = x+2$, valid for every $x \neq 2$.
3. Now substitute: $2 + 2 = 4$.

The answer is 4: near the hole in the graph, the function hugs the line $x + 2$. The moral — simplify first, substitute second — carries the whole technique. Hand the AI box on this calculus solver page any 0/0 limit and watch it run these three moves.

1. Treating a limit as substitution. Plugging in first only works when nothing breaks. When substitution yields $\dfrac{0}{0}$, simplify — factor and cancel as in $\lim_{x\to2}\dfrac{x^2-4}{x-2}$ — and substitute after. Order matters: simplify first, substitute second.
2. Forgetting the chain rule. A function inside a function differentiates to the outer derivative times the inner derivative. Leave out that inner factor and the answer is incomplete — the most common lost point on derivative questions.
3. Integrating without the + C. Every indefinite integral ends in $+ C$, because differentiation erases constants and the return trip cannot rebuild them. Dropping it reports one curve from a family of infinitely many.

The calculus solver above applies the chain rule every time it should — and the AI box shows its steps, so you can compare line by line.