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Differentiation practice

QCE Mathematical Methods · Original practice questions with worked solutions

All differentiation practice questions

98 original questions · Page 3 of 5

  1. Q156 · Original practice · 1 mark
    Determine ddx[cos⁡(5x)]\dfrac{d}{dx}[\cos(5x)].
    Differentiation of trigonometric functions and differentiation rules
  2. Q157 · Original practice · 1 mark
    For g(x)=xsin⁡xg(x)=x\sin x, determine g′(π)g'(\pi).
    Differentiation of trigonometric functions and differentiation rules
  3. Q158 · Original practice · 1 mark
    The graph of y=sin⁡(2x)y=\sin(2x) is shown with the point x=π8x=\dfrac{\pi}{8} marked. The gradient of the tangent at this point is
    Differentiation of trigonometric functions and differentiation rules
  4. Q159 · Original practice · 4 marks
    For h(x)=ln⁡xx+1h(x)=\dfrac{\ln x}{x+1}, x>0x>0, determine the equation of the tangent to the graph of hh at x=1x=1.
    Differentiation of trigonometric functions and differentiation rules
  5. Q160 · Original practice · 5 marks
    Let f(x)=sin⁡x2+cos⁡xf(x)=\dfrac{\sin x}{2+\cos x} for 0≤x≤2π0\le x\le2\pi. Determine the global maximum and global minimum values of ff on this interval, and the xx-values at which they occur.
    Differentiation of trigonometric functions and differentiation rules
  6. Q161 · Original practice · 1 mark
    A differentiable function has a stationary point at x=3x=3 and f′′(3)>0f''(3)>0. The stationary point is
    Further applications of differentiation
  7. Q162 · Original practice · 1 mark
    If f′′(x)=6x−12f''(x)=6x-12, the point of inflection occurs at
    Further applications of differentiation
  8. Q163 · Original practice · 1 mark
    A particle has displacement s(t)=t3−6t2+9ts(t)=t^3-6t^2+9t metres. Its acceleration at t=2t=2 seconds is
    Further applications of differentiation
  9. Q164 · Original practice · 5 marks
    For f(x)=x3−6x2+9x+1f(x)=x^3-6x^2+9x+1, determine the coordinates and nature of all stationary points and determine the point of inflection.
    Further applications of differentiation
  10. Q165 · Original practice · 6 marks
    An open-top box with a square base has volume 500 cm3500\text{ cm}^3. Let the base side length be xx cm and the height be hh cm, as shown. Determine the dimensions that minimise the amount of material required to make the box.
    Further applications of differentiation
  11. Q201 · Original practice · 1 mark
    For f(x)=x2ex,f′(x)f(x)=x^{2} e^{x}, f'(x) is
    Differentiation of exponential and logarithmic functions
  12. Q212 · Original practice · 5 marks
    Let f(x)=x3−6x2+9x+4f(x)=x^{3}-6x^{2}+9x+4.
    Further applications of differentiation
  13. Q217 · Original practice · 6 marks
    A rectangular poster has area 600 cm2600\,\mathrm{cm}^{2}. It has 2 cm2\,\mathrm{cm} margins on the left and right and 3 cm3\,\mathrm{cm} margins at the top and bottom. Let the poster width be wcmw cm.
    Further applications of differentiation
  14. Q222 · Original practice · 1 mark
    ddxln⁡(3x2+1)\frac{d}{\mathrm{d}x} \ln (3x^{2}+1) equals
    Differentiation of exponential and logarithmic functions
  15. Q229 · Original practice · 1 mark
    For f(x)=x3−3xf(x)=x^{3}-3x, a stationary point occurs at
    Further applications of differentiation
  16. Q237 · Original practice · 6 marks
    A closed cylindrical can must hold 750 cm3750\,\mathrm{cm}^{3}. Let rr be its radius.
    Further applications of differentiation
  17. Q239 · Original practice · 1 mark
    A lantern display has brightness B(t)=120e−0.3tB(t)=120e^{-0.3t}, where tt is measured in minutes. Which expression gives B′(t)B^{\prime}(t)?
    Differentiation of exponential and logarithmic functions
  18. Q240 · Original practice · 1 mark
    The graph represents y=ln⁡(x−2)+1y=\ln(x-2)+1. Which pair states its domain and vertical asymptote?
    Differentiation of exponential and logarithmic functions
  19. Q241 · Original practice · 1 mark
    For f(x)=x2sin⁡xf(x)=x^2\sin x, the derivative is
    Differentiation of trigonometric functions and differentiation rules
  20. Q242 · Original practice · 1 mark
    If y=cos⁡(3x−1)y=\cos(3x-1), then dydx\dfrac{dy}{dx} is
    Differentiation of trigonometric functions and differentiation rules
  21. Q243 · Original practice · 1 mark
    The graph shows f′(x)=(x+2)(x−1)f^{\prime}(x)=(x+2)(x-1). At which value of xx does ff have a local maximum?
    Further applications of differentiation
  22. Q244 · Original practice · 1 mark
    For f(x)=x3−6x2+9x+4f(x)=x^3-6x^2+9x+4, the point of inflection is
    Further applications of differentiation
  23. Q267 · Original practice · 1 mark
    For f(x)=ln⁡xxf(x)=\dfrac{\ln x}{x}, where x>0x>0, f′(x)f^{\prime}(x) is
    Differentiation of trigonometric functions and differentiation rules
  24. Q269 · Original practice · 3 marks
    Consider the equation ln⁡(x−1)+ln⁡(x+1)=ln⁡8\ln(x-1)+\ln(x+1)=\ln8.
    Differentiation of exponential and logarithmic functions