Skip to main content
EasyPalmCards

Flash card deck22 cardsIntermediateUniversity

Calculus basics: derivatives flash cards

Flash cards on derivatives: what a derivative means, the power, product, quotient and chain rules, derivatives of common functions, and stationary points.

Use this template Study now

Free to use, edit, print and study. No account needed: your set stays on this device.

The cards, front and back

  1. 1
    Front

    What does a derivative measure?

    Back

    The instantaneous rate of change of a function, which is the gradient of its graph at a point. If y = f(x), the derivative is written dy/dx or f′(x).

  2. 2
    Front

    What is the definition of the derivative from first principles?

    Back

    f′(x) is the limit, as h tends to 0, of (f(x + h) − f(x)) ÷ h.

  3. 3
    Front

    Derivative of a constant

    Back

    The derivative of any constant is 0, because its graph is a flat line.

  4. 4
    Front

    The power rule

    Back

    The derivative of x^n is nx^(n − 1), for any real number n. For example, the derivative of x^5 is 5x^4.

  5. 5
    Front

    Derivative of √x

    Back

    √x = x^(1/2), so its derivative is ½x^(−1/2) = 1 ÷ (2√x).

  6. 6
    Front

    Derivative of 1/x

    Back

    1/x = x^(−1), so its derivative is −x^(−2) = −1 ÷ x^2.

  7. 7
    Front

    Constant multiple and sum rules

    Back

    The derivative of k f(x) is k f′(x). The derivative of f(x) + g(x) is f′(x) + g′(x), so you can differentiate a polynomial term by term.

  8. 8
    Front

    Differentiate 3x^4 − 2x^2 + 7x − 5.

    Back

    12x^3 − 4x + 7.

  9. 9
    Front

    Derivative of sin x

    Back

    cos x (with x in radians).

  10. 10
    Front

    Derivative of cos x

    Back

    −sin x (with x in radians).

  11. 11
    Front

    Derivative of tan x

    Back

    sec^2 x, which equals 1 ÷ cos^2 x (with x in radians).

  12. 12
    Front

    Derivative of e^x

    Back

    e^x. It is the only function (apart from multiples of it) that is its own derivative.

  13. 13
    Front

    Derivative of e^(kx)

    Back

    ke^(kx), using the chain rule.

  14. 14
    Front

    Derivative of ln x

    Back

    1/x, for x greater than 0.

  15. 15
    Front

    Derivative of a^x

    Back

    a^x ln a, for a positive constant a.

  16. 16
    Front

    The chain rule

    Back

    If y = f(g(x)), then dy/dx = f′(g(x)) × g′(x). Differentiate the outer function, keep the inner function, then multiply by the derivative of the inner function.

  17. 17
    Front

    Differentiate (2x + 1)^5.

    Back

    5(2x + 1)^4 × 2 = 10(2x + 1)^4.

  18. 18
    Front

    The product rule

    Back

    If y = uv, then dy/dx = u′v + uv′.

  19. 19
    Front

    The quotient rule

    Back

    If y = u ÷ v, then dy/dx = (u′v − uv′) ÷ v^2.

  20. 20
    Front

    What is a stationary point?

    Back

    A point where the gradient is zero, f′(x) = 0. It can be a local maximum, a local minimum or a point of inflection.

  21. 21
    Front

    How does the second derivative test work?

    Back

    At a stationary point, if f″(x) is greater than 0 it is a local minimum, and if f″(x) is less than 0 it is a local maximum. If f″(x) = 0, the test does not decide and you need another method.

  22. 22
    Front

    How do you find the equation of a tangent at x = a?

    Back

    Find the gradient m = f′(a) and the point (a, f(a)), then use y − f(a) = m(x − a).

How to use this template

This deck covers the first part of differential calculus: what a derivative means, the definition from first principles, the power, constant multiple and sum rules, the derivatives of common functions (powers, roots, reciprocals, sin, cos, tan, e^x, ln x and a^x), the chain, product and quotient rules with worked examples, stationary points, the second derivative test and tangent lines. It suits first-year university calculus, A level and IB maths, AP Calculus AB, Year 11 and Year 12 maths methods courses and anyone refreshing calculus before an engineering, science or economics degree. Drill the standard derivatives in Test mode until they are automatic, let spaced repetition handle the rules, and print the deck double-sided as a reference.

Tips

  • Rewrite roots and fractions as powers before using the power rule, for example 1/x^2 as x^(−2).
  • Say the chain rule aloud as you use it: derivative of the outside, keep the inside, times derivative of the inside.
  • Remember that the trig derivatives only work in radians. Switch your calculator to radian mode for calculus.
  • Check signs carefully with cos x and in the quotient rule, where the order of u′v − uv′ matters.
  • After finding a derivative, check it quickly by testing the gradient at one point on a graphing tool.

Make it fit

  • Add a matching set of integration cards with each derivative reversed.
  • Add implicit differentiation, inverse trig functions or partial derivatives for a later course.
  • Make reverse cards with the derivative on the front and the original function on the back.
  • Use the notation your course prefers, dy/dx or f′(x), and keep it consistent across the deck.

Updated October 28, 2026

Browse all templates