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Calculus

Differentiate a polynomial with the power rule, find the slope at a point, graph the tangent line, then integrate — rules, meaning and pictures together.

7 min read Students · Educators Video in 1 language
2:27 · narrated · captions

Set up a calculus session

  1. Open the Setup Wizard

    In the tutor lobby, choose Setup Wizard at the top (or Customize). The wizard builds a focused session in four quick steps.

  2. Pick your level — Expert

    Build on Basics revisits foundations, Advanced (recommended) works at grade level, and Expert moves faster with harder problems. Then press Continue.

  3. Choose how you learn — In-depth

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  4. Choose Math, then the focus area — Calculus

    Select Math. A Focus area in Math row appears (General Math, Arithmetic, Algebra, Geometry, Trigonometry, Pre-Calculus, Calculus, Statistics, Probability and more). Picking one keeps every explanation, example and practice question on that topic.

  5. Pick a buddy and start

    Choose a learning buddy — Sage is calm and methodical, a favourite for math — then press Start Session.

Derivatives and slope

f(x) = 3x² + 2x − 5: f′(x), f′(1), ∫(6x + 2) dxGrades 11–12 · college
  1. Power ruled/dx[3x²] = 6x, d/dx[2x] = 2, d/dx[−5] = 0d/dx[xⁿ] = n·xⁿ⁻¹; constants vanish.
  2. Derivativef′(x) = 6x + 2Add the term derivatives.
  3. Slope at 1f′(1) = 6(1) + 2 = 8Tangent line: y = 8x − 8 (through (1, 0)).
  4. Integral∫(6x + 2) dx = 3x² + 2x + CReverse power rule: ∫xⁿ dx = xⁿ⁺¹/(n+1).
Answer: f′(x) = 6x + 2, slope 8 at x = 1, ∫ = 3x² + 2x + C
Check: f(1) = 3 + 2 − 5 = 0, so the tangent passes through (1, 0) ✓

  1. Ask for the derivative

    Write functions as f(x) = … and powers with ^.

    Try it“Find the derivative of f(x) = 3x^2 + 2x - 5”

  2. Find the slope at a point

    The derivative is the slope. Plug in x = 1.

    Try it“What is the slope of the curve at x = 1?”

  3. Graph the tangent line

    See the curve and the line that just touches it at x = 1.

    Try it“Can you graph f(x) and its tangent line at x = 1?”

  4. Integrate back

    Integration reverses differentiation — up to a constant.

    Try it“Now integrate 6x + 2 with respect to x.”

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