Socratic Solve
The short answer
A calculus solver with steps shows how the answer was reached: the rule applied on each line, what it does to the expression, and a check at the end, such as differentiating an antiderivative to get the integrand back. Whether you call it a calculus problem solver or a derivative and integral solver with steps, that check is what separates a useful tool from an answer printer. You can scan the problem with your camera, type it, or upload a photo, and Socratic Solve returns the working line by line in rendered math notation.
Below are five worked examples covering limits, derivatives, integrals and a word problem, each ending with a check.
Five worked calculus examples, with steps
Try each one on paper first, then compare your lines to these.
1. Limits: factoring and a standard limit
(a) lim as x → 2 of (x² − 4) / (x − 2)
- Try direct substitution: (4 − 4) / (2 − 2) = 0/0. That's indeterminate, not "undefined", so you need another route.
- Factor the numerator (difference of squares): x² − 4 = (x − 2)(x + 2).
- Cancel the common factor: for x ≠ 2, (x − 2)(x + 2) / (x − 2) = x + 2.
- Now substitute: 2 + 2 = 4.
Check: plug in a value close to 2. At x = 2.01: (4.0401 − 4) / 0.01 = 4.01. The outputs approach 4, so the limit is 4.
(b) lim as x → 0 of sin(5x) / x
- Substitution gives 0/0 again.
- Use the standard limit lim as θ → 0 of sin(θ)/θ = 1. Rewrite: sin(5x)/x = 5 · sin(5x)/(5x).
- Let θ = 5x. As x → 0, θ → 0, so the limit is 5 · 1 = 5.
Check with L'Hôpital's rule: differentiate top and bottom separately: 5cos(5x) / 1, which is 5 at x = 0. Numerically, sin(0.05)/0.01 ≈ 4.998. All three agree: the limit is 5.
2. Derivatives: product rule and chain rule
(a) Product rule: d/dx [x² sin(3x)]
- Identify the two factors: u = x² and v = sin(3x).
- Differentiate each: u′ = 2x. For v, use the chain rule: v′ = cos(3x) · 3 = 3cos(3x).
- Apply (uv)′ = u′v + uv′: 2x sin(3x) + x² · 3cos(3x).
- Tidy up: 2x sin(3x) + 3x² cos(3x).
Check: at x = 1 the formula gives 2sin(3) + 3cos(3) ≈ −2.6877 (calculator in radians). A difference quotient, (f(1.001) − f(0.999)) / 0.002, gives the same −2.6877.
(b) Chain rule: d/dx [(3x² + 1)⁵]
- Outer function: something to the 5th power. Its derivative is 5(something)⁴.
- Inner function: 3x² + 1, with derivative 6x.
- Multiply outer derivative by inner derivative: 5(3x² + 1)⁴ · 6x.
- Simplify: 30x(3x² + 1)⁴.
Check: at x = 1, 30 · 1 · 4⁴ = 7,680. The difference quotient at x = 1 gives about 7,680.06, which matches.
3. Integral by u-substitution: ∫ x(x² + 1)⁴ dx
- Pick u as the inside of the power: u = x² + 1.
- Find du: du = 2x dx, so x dx = du/2.
- Substitute: ∫ u⁴ · (du/2) = (1/2) ∫ u⁴ du.
- Integrate: (1/2) · u⁵/5 = u⁵/10.
- Substitute back and add the constant: (x² + 1)⁵/10 + C.
Check by differentiating: d/dx [(x² + 1)⁵/10] = 5(x² + 1)⁴ · 2x / 10 = x(x² + 1)⁴. That's the original integrand, so the antiderivative is right. This check works on every indefinite integral.
4. Definite integral: ∫₀¹ 2x(x² + 1)³ dx
- Substitute u = x² + 1, so du = 2x dx and the integrand becomes u³ du.
- Change the limits to u values: when x = 0, u = 1; when x = 1, u = 2.
- Integrate: ∫₁² u³ du = [u⁴/4] from 1 to 2.
- Evaluate: 16/4 − 1/4 = 15/4 = 3.75.
Check: go back to x instead of changing limits. The antiderivative is (x² + 1)⁴/4, and (2⁴ − 1⁴)/4 = 15/4. Same value. A definite integral has no + C, because it cancels when you subtract.
5. Optimization word problem: fencing along a river
A farmer has 200 m of fencing to build a rectangular pen along a straight river. No fence is needed on the river side. What dimensions give the largest area?
- Name the variables: let x be each side perpendicular to the river. The side parallel to the river is then 200 − 2x.
- Write the quantity to maximize: A(x) = x(200 − 2x) = 200x − 2x², with 0 < x < 100.
- Differentiate and set to zero: A′(x) = 200 − 4x = 0, so x = 50.
- Confirm it's a maximum: A″(x) = −4, which is negative, so x = 50 is a maximum.
- Answer the question asked: the pen is 50 m by 100 m, with an area of 5,000 m².
Check: try nearby values. x = 49 gives 49 × 102 = 4,998 m², and x = 51 gives 51 × 98 = 4,998 m². Both are smaller than 5,000, so 50 m is the best choice. Note the last step: the question asked for dimensions, so "x = 50" alone would lose a mark.
Solve your calculus problem, with the working
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The calculus mistakes that cost the most marks
Most lost calculus marks come from a short list of repeat errors.
| Mistake | Wrong | Right |
|---|---|---|
| Forgetting the chain rule's inner derivative | d/dx sin(3x) = cos(3x) | d/dx sin(3x) = 3cos(3x) |
| Dropping + C | ∫ 2x dx = x² | ∫ 2x dx = x² + C |
| Sign errors with trig | d/dx cos x = sin x | d/dx cos x = −sin x, and ∫ sin x dx = −cos x + C |
| Not changing limits in u-substitution | ∫₀¹ 2x(x² + 1)³ dx = [u⁴/4] from 0 to 1 = 1/4 | Limits become 1 to 2: [u⁴/4] from 1 to 2 = 15/4 |
| Product rule as product of derivatives | d/dx (x² sin x) = 2x cos x | d/dx (x² sin x) = 2x sin x + x² cos x |
| Calculator in degree mode | sin(0.01)/0.01 ≈ 0.0175 in degrees | In radians it's ≈ 1.0000, matching lim sin(x)/x = 1 |
The last one catches strong students. Every trig derivative rule assumes radians. In degree mode, sin(x)/x near zero approaches π/180 ≈ 0.01745 instead of 1, and every numeric check you do will look wrong. Set your calculator to radians before a calculus test.
How to photograph or type calculus so a solver reads it
- Make the bounds readable. Write the upper and lower limits of an integral clearly above and below the ∫ sign, not squeezed against it.
- Always include dx. It tells the solver (and your teacher) which variable you're integrating with respect to, and where the integrand ends.
- Keep exponents distinct. Write powers small and high so x² doesn't read as x2, and use brackets for compound powers like (3x² + 1)⁵.
- Use parentheses for function arguments. sin(3x) is unambiguous; sin 3x can be read as (sin 3)·x.
- Check the recognized problem against your page before reading the solution. One misread exponent changes everything after it.
Typing works well for short problems. Plain phrasing with parentheses is enough: "integral of x^2 from 0 to 3", "derivative of x^2 sin(3x)", or "limit of (x^2-4)/(x-2) as x approaches 2".
How to read a step-by-step calculus solution
- Attempt it first. Even a few lines show you where you got stuck, and that line is the one to study.
- Name the rule on each line. "Chain rule, inner derivative 6x." "Substituted u = x² + 1." "Changed limits." If you can't name a step, that's your gap.
- Find the first line where you differ. In calculus it's usually a missing inner derivative or a sign, and everything after it is correct method on wrong numbers.
- Cover the solution and redo it. Solve from a blank page, then compare.
- Do a similar problem with new numbers. Swap sin(3x) for cos(4x), or 200 m of fencing for 360 m. If you can do the variation without help, you've learned the method.
- Run the check yourself. Differentiate the antiderivative, test a nearby value, or plug in numbers. If a solution doesn't include a check, add one.
Using Socratic Solve as your calculus solver with steps
Socratic Solve covers calculus alongside algebra, geometry, trig, logarithms and linear algebra, plus physics, chemistry, biology and more. Scan a printed or handwritten problem, type it, or upload a photo, and the solution comes back step by step in real rendered notation. Every problem is saved to your History, tagged by subject and searchable, so you can line up five chain rule problems and spot the pattern. It's on iPhone, iPad and Android, with free credits to start.
If your calculus is shaky because the algebra underneath it is, start with the algebra solver with steps guide, since factoring and exponents show up on almost every line of calculus. For a side-by-side look at different apps, see the best homework helper apps.