d/dx

Derivatives: Rules, Techniques & Applications

136 cards · 🇺🇸 English

Differentiation rules and techniques, plus applications like optimization and related rates.

calculusderivatives

Study all 136 cards for free

Flashcards, quizzes, pixel-art games and a daily review that remembers for you.

Study this deck free

Try the first 10 cards

1 / 10
Question

Derivative

Knew itLearning

Tap the card for the answer · swipe right if you knew it, left if not

Already have an account? Sign in

Cards

  1. 1Derivative
    A fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input; the slope of the tangent line to the graph at a point
  2. 2What does the derivative represent geometrically?
    The slope of the tangent line to the graph of the function at a given point
  3. 3What does the derivative represent physically?
    The instantaneous rate of change; the ratio of the instantaneous change in the dependent variable to that of the independent variable
  4. 4Differentiation
    The process of finding a derivative
  5. 5Definition of a derivative using limits
    f'(a) = lim[h→0] (f(a+h) - f(a))/h, provided the limit exists
  6. 6Leibniz notation for the derivative
    dy/dx or df/dx, representing the derivative as a ratio of differentials
  7. 7Prime notation for the derivative
    f'(x) (read 'f prime of x'), introduced by Lagrange
  8. 8Newton's dot notation for derivatives
    ẋ for the first derivative and ẍ for the second derivative with respect to time
  9. 9D-notation for derivatives
    Df or D^n f for the nth derivative, using the differential operator D
  10. 10If f is differentiable at a point, must it be continuous there?
    Yes. Differentiability implies continuity
  11. 11If f is continuous at a point, must it be differentiable there?
    No. Continuous functions may not be differentiable (e.g., |x| at x = 0)
  12. 12Derivative of a constant function
    Zero. If f(x) = c, then f'(x) = 0
  13. 13Power rule for derivatives
    If f(x) = x^n, then f'(x) = nx^(n-1)
  14. 14Find d/dx[x³]
    3x². Using the power rule: bring down the exponent and subtract 1
  15. 15Find d/dx[x⁵]
    5x⁴. Using the power rule
  16. 16Find d/dx[1/x]
    -1/x². Rewrite as x^(-1), then apply power rule: -1·x^(-2) = -1/x²
  17. 17Find d/dx[√x]
    1/(2√x). Rewrite as x^(1/2), then (1/2)x^(-1/2) = 1/(2√x)
  18. 18Constant multiple rule
    d/dx[cf(x)] = c·f'(x), where c is a constant
  19. 19Sum rule for derivatives
    d/dx[f(x) + g(x)] = f'(x) + g'(x). The derivative of a sum is the sum of derivatives
  20. 20Difference rule for derivatives
    d/dx[f(x) - g(x)] = f'(x) - g'(x)
  21. 21Find d/dx[3x⁴ + 5x² - 7]
    12x³ + 10x. Apply sum rule and power rule to each term
  22. 22Product rule
    d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x). (First·derivative of second + second·derivative of first)
  23. 23Find d/dx[x²·sin(x)] using the product rule
    2x·sin(x) + x²·cos(x)
  24. 24Quotient rule
    d/dx[f(x)/g(x)] = [f'(x)g(x) - f(x)g'(x)]/[g(x)]², where g(x) ≠ 0
  25. 25Mnemonic for the quotient rule
    'Low D-high minus high D-low, over low squared' (denominator times derivative of numerator minus numerator times derivative of denominator, all over denominator squared)
  26. 26Find d/dx[x/cos(x)] using the quotient rule
    [cos(x) - x·(-sin(x))]/cos²(x) = [cos(x) + x·sin(x)]/cos²(x)
  27. 27Chain rule
    If y = f(g(x)), then dy/dx = f'(g(x))·g'(x). The derivative of a composite function equals the derivative of the outer function times the derivative of the inner function
  28. 28Find d/dx[(3x + 2)⁵] using the chain rule
    5(3x + 2)⁴·3 = 15(3x + 2)⁴
  29. 29Find d/dx[sin(x²)]
    cos(x²)·2x = 2x·cos(x²). Apply chain rule
  30. 30Derivative of sin(x)
    cos(x)

And 106 more cards. Study the whole deck free

Articles about this deck

More Mathematics decks

See all
Study all 136 cards free