SolveMathAI

Calculus solver

Limits, derivatives, integrals and the word problems built on them.

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  • Arithmetic to calculus

Calculus is two ideas and a bridge between them. The derivative measures how fast something changes; the integral adds up infinitely many small pieces; the fundamental theorem says each undoes the other. Nearly every exam question is one of those, dressed up in a context.

Naming the question first

Most lost marks come from starting the wrong machine, not from bad arithmetic. Identify the shape before you differentiate anything.

  1. Rate of change, slope, tangent, velocity

    That is a derivative. Decide which rule applies: product, quotient or chain, and often more than one.

  2. Area, total, accumulated, displacement, average value

    That is an integral. Definite if it has limits, indefinite if it needs a + C.

  3. Behaviour near a point, or at infinity

    That is a limit. Substitute first; only reach for factoring, conjugates or L’Hôpital when substitution gives 0/0 or ∞/∞.

  4. Two quantities changing together

    That is related rates. Write the equation connecting them, differentiate both sides with respect to time, then substitute the numbers last.

  5. Maximum or minimum

    That is optimisation. Set the derivative to zero, solve, and test whether each point is a maximum or a minimum.

Worked examples

Each one is solved the way the solver solves it: the rule first, then the line.

Find the derivative of (2x + 1)^5

ddx (2x+1)5\frac{d}{dx}\,(2x + 1)^5
  1. Chain rule: outside first
    5(2x+1)4⋅ddx(2x+1)5(2x + 1)^4 \cdot \frac{d}{dx}(2x + 1)
  2. Derivative of the inside
    ddx(2x+1)=2\frac{d}{dx}(2x + 1) = 2
  3. Multiply
    10(2x+1)410(2x + 1)^4
Answer
10(2x+1)410(2x + 1)^4

Evaluate the integral of (2x + 1) dx from 0 to 3

∫03(2x+1) dx\int_0^3 (2x + 1)\,dx
  1. Antiderivative term by term
    ∫(2x+1) dx=x2+x\int (2x + 1)\,dx = x^2 + x
  2. Fundamental theorem of calculus
    [x2+x]03=(9+3)−(0+0)\Big[ x^2 + x \Big]_0^3 = (9 + 3) - (0 + 0)
  3. Evaluate
    1212
Answer
1212

Find the limit as x approaches 0 of sin(x)/x

lim⁡x→0sin⁡xx\lim_{x \to 0} \frac{\sin x}{x}
  1. Substitution gives an indeterminate form
    sin⁡00=00\frac{\sin 0}{0} = \frac{0}{0}
  2. This is a standard limit
    lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1
  3. Or by L’Hôpital’s rule
    lim⁡x→0cos⁡x1=cos⁡0=1\lim_{x \to 0} \frac{\cos x}{1} = \cos 0 = 1
Answer
11

A 10 ft ladder leans against a wall. The bottom slides away at 2 ft/s. How fast is the top sliding down when the bottom is 6 ft from the wall?

x2+y2=102x^2 + y^2 = 10^2
  1. Relate the quantities
    x2+y2=100x^2 + y^2 = 100
  2. Differentiate both sides with respect to t
    2xdxdt+2ydydt=02x \frac{dx}{dt} + 2y \frac{dy}{dt} = 0
  3. Find y when x = 6
    y=100−36=8y = \sqrt{100 - 36} = 8
  4. Substitute the known values
    2(6)(2)+2(8)dydt=02(6)(2) + 2(8)\frac{dy}{dt} = 0
  5. Solve
    dydt=−2416=−1.5\frac{dy}{dt} = -\frac{24}{16} = -1.5
Answer
dydt=−1.5 ft/s\frac{dy}{dt} = -1.5\ \text{ft/s}

The minus sign is the answer to "how fast is it falling": the height is decreasing at 1.5 ft/s.

Where marks get lost

Substituting numbers too early in related rates

Differentiate the general relationship first, then put the numbers in. Substituting first freezes the very quantity that is supposed to be changing.

Losing the + C

An indefinite integral is a family of functions. Without + C the answer is one member of it, and in a differential equation that costs the whole question.

Reaching for L’Hôpital too fast

It only applies to 0/0 and ∞/∞. Used on anything else it produces a confident wrong answer.

Questions

What level of calculus does it handle?

AP Calculus AB and BC, and first-year university single-variable calculus: limits, continuity, derivatives, applications, integrals, techniques of integration, sequences and series. It also handles common multivariable topics like partial derivatives and double integrals.

Does it show which rule was used?

Yes. Each line names the rule, so you can see exactly where the chain rule came in rather than staring at a finished answer.

Can it do proofs or epsilon-delta arguments?

It attempts them, and it is weaker there than on computation. Treat a proof as a draft to check, not as an answer.

Are the integrals exact?

Yes where an exact form exists, with the constant of integration included. When an integral has no elementary antiderivative it says so and gives a numerical value instead.