HexMathv1.0.3

Calculus solver

Type or scan a limit, derivative or integral. HexMath solves it step by step.

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What this page covers

  • Limits, including ones that start as 00\frac{0}{0} and limits at infinity.
  • Derivatives with the product, quotient and chain rules.
  • Indefinite and definite integrals of standard functions.
  • Not covered here: differential equations and transforms; for longer derivative or integral work see the derivative calculator and the integral calculator.

How to enter the problem

  • Type it or paste it. Write lim, d/dx or the integral sign with its limits, for example lim x->0 sin(3x)/x.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Differentiate y=x3sin⁡x\displaystyle y = x^3 \sin x.

Answer

Verified

The answer is

dydx=3x2sin⁡x+x3cos⁡x\frac{dy}{dx} = 3x^2 \sin x + x^3 \cos x

Explanation

  1. Identify the structure
    This is a product of u=x3u = x^3 and v=sin⁡xv = \sin x, so use the product rule (uv)′=u′v+uv′(uv)' = u'v + uv'.
  2. Differentiate each factor
    u′=3x2u' = 3x^2 and v′=cos⁡xv' = \cos x.
  3. Combine
    y′=3x2sin⁡x+x3cos⁡xy' = 3x^2 \sin x + x^3 \cos x

Example 2

Problem

Evaluate ∫02(3x2−2x) dx\displaystyle \int_0^2 (3x^2 - 2x)\,dx.

Answer

Verified

The answer is

44

Explanation

  1. Find an antiderivative
    ∫(3x2−2x) dx=x3−x2\int (3x^2 - 2x)\,dx = x^3 - x^2
  2. Evaluate at the limits
    At x=2x = 2: 8−4=48 - 4 = 4. At x=0x = 0: 00.
  3. Subtract
    4−0=44 - 0 = 4

Example 3

Problem

Find lim⁡x→∞2x2+15x2−x\displaystyle \lim_{x \to \infty} \frac{2x^2 + 1}{5x^2 - x}.

Answer

Verified

The answer is

25\frac{2}{5}

Explanation

  1. Divide by the highest power
    Divide top and bottom by x2x^2: 2+1x25−1x\frac{2 + \frac{1}{x^2}}{5 - \frac{1}{x}}.
  2. Let xx grow
    1x2\frac{1}{x^2} and 1x\frac{1}{x} both go to 00.
  3. Read off the limit
    2+05−0=25\frac{2 + 0}{5 - 0} = \frac{2}{5}

Common mistakes

  • Cancelling inside the sine: sin⁡3xx\frac{\sin 3x}{x} is not sin⁡3\sin 3. The limit is 33.
  • Multiplying derivatives in a product: (x3sin⁡x)′(x^3 \sin x)' is not 3x2cos⁡x3x^2 \cos x.
  • Forgetting to subtract the value at the lower limit in a definite integral.
  • Comparing the wrong terms at infinity. Only the highest powers decide the limit, so it is 25\frac{2}{5}, not 15\frac{1}{5} or 00.

Checks, assumptions and limits

  • Check by hand: put a value close to the limit point into the expression, such as x=0.001x = 0.001 for the example, and see that the result is close to 33.
  • Angles in calculus are in radians. Limits and derivatives of trig functions change if the angle is in degrees.
  • HexMath can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Limits, derivatives and integrals, both indefinite and definite. Type the problem or scan it from a page.

Yes. Each step has a short title, such as the product rule or evaluating at the limits, then the working.

Yes. Ask in the same thread, for example why a rule applies. A typed follow-up needs you to be signed in.

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