Derivatives of Exponential, Logarithmic and Trigonometric Functions · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Derivatives of e, logs and trig

Mathematics · Calculus & Analysis · ages 18-19
Name ______________________   Date ____________

  1. With x measured in radians, what is the derivative of sin x?

    • cos x
    • -sin x
    • -cos x

  2. What is the derivative of e^(3x)?

    • e^(3x)
    • 3x times e^(3x - 1)
    • 3e^(3x)

  3. For x > 0, what is the derivative of ln(5x)?

    • 5/x
    • 1/x
    • 1/(5x)

  4. The derivative of e^x is e^x.

    Circle one:   True   False

  5. Which choice gives both the derivative of ln(4-2x) and its real domain?

    • 1/(4-2x), for x < 2
    • -2/(4-2x), for x > 2
    • -2/(4-2x), for x < 2

  6. The gradient of y = cos x is -sin x. Use sin x ≈ x near zero to estimate this gradient at x = 0.04 radians.

    Answer: ______________

  7. Differentiate ln(x^2+4) for real x.

    • 1/(x^2+4)
    • 2x/(x^2+4)
    • 2x/(x^2+4)^2

  8. For y = sin x in radians, what describes its tangent at x = pi/2?

    • Slope 1, so the tangent rises one unit per unit of x.
    • Slope -1, so the tangent falls one unit per unit of x.
    • Slope 0, so the tangent is horizontal.

  9. A curve y = a^x, with a > 0, has height 5 and gradient 2 at a point. Which equation determines its base?

    • ln a = 5/2
    • ln a = 2/5
    • ln a = 2 - 5

  10. What is the derivative of 2^x?

    • x times 2^(x - 1)
    • 2^x ln 2
    • 2^x
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Answer key

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Derivatives of e, logs and trig W1-mt_695WPZKADT-s1

  1. cos x · The radian-based sine derivative is cos x. Negative sine is the derivative of cosine.
  2. 3e^(3x) · Keep e^(3x) from the outer rule, then multiply by the inner derivative 3.
  3. 1/x · Split into ln 5 + ln x. The constant vanishes and ln x gives 1/x.
  4. True · That is what makes e special: the ln factor equals 1 and disappears.
  5. -2/(4-2x), for x < 2 · A positive argument requires 4-2x > 0, so x < 2. The argument has derivative -2, giving -2/(4-2x).
  6. -0.04 · The sine tangent gives sin(0.04) ≈ 0.04. Retain the cosine derivative's minus sign to obtain -0.04.
  7. 2x/(x^2+4) · The argument x^2+4 is positive for every real x. Divide its derivative 2x by the original argument.
  8. Slope 0, so the tangent is horizontal. · The derivative is cos x. At pi/2, cos x equals 0, giving a horizontal tangent.
  9. ln a = 2/5 · For a^x, gradient divided by height equals ln a. Here the ratio is 2/5.
  10. 2^x ln 2 · For a > 0, (a^x)' = a^x ln a. With a = 2, the derivative is 2^x ln 2.
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