Pauls Online Notes
Pauls Online Notes
Home / Calculus I / Derivatives / Differentiation Formulas
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

Section 3-3 : Differentiation Formulas

For problems 1 – 12 find the derivative of the given function.

  1. \(f\left( x \right) = 6{x^3} - 9x + 4\) Solution
  2. \(y = 2{t^4} - 10{t^2} + 13t\) Solution
  3. \(g\left( z \right) = 4{z^7} - 3{z^{ - 7}} + 9z\) Solution
  4. \(h\left( y \right) = {y^{ - 4}} - 9{y^{ - 3}} + 8{y^{ - 2}} + 12\) Solution
  5. \(y = \sqrt x + 8\,\sqrt[3]{x} - 2\,\sqrt[4]{x}\) Solution
  6. \(f\left( x \right) = 10\,\sqrt[5]{{{x^3}}} - \sqrt {{x^7}} + 6\,\sqrt[3]{{{x^8}}} - 3\) Solution
  7. \(\displaystyle f\left( t \right) = \frac{4}{t} - \frac{1}{{6{t^3}}} + \frac{8}{{{t^5}}}\) Solution
  8. \(\displaystyle R\left( z \right) = \frac{6}{{\sqrt {{z^3}} }} + \frac{1}{{8{z^4}}} - \frac{1}{{3{z^{10}}}}\) Solution
  9. \(z = x\left( {3{x^2} - 9} \right)\) Solution
  10. \(g\left( y \right) = \left( {y - 4} \right)\left( {2y + {y^2}} \right)\) Solution
  11. \(\displaystyle h\left( x \right) = \frac{{4{x^3} - 7x + 8}}{x}\) Solution
  12. \(\displaystyle f\left( y \right) = \frac{{{y^5} - 5{y^3} + 2y}}{{{y^3}}}\) Solution
  13. Determine where, if anywhere, the function \(f\left( x \right) = {x^3} + 9{x^2} - 48x + 2\) is not changing. Solution
  14. Determine where, if anywhere, the function \(y = 2{z^4} - {z^3} - 3{z^2}\) is not changing. Solution
  15. Find the tangent line to \(\displaystyle g\left( x \right) = \frac{{16}}{x} - 4\sqrt x \) at \(x = 4\). Solution
  16. Find the tangent line to \(f\left( x \right) = 7{x^4} + 8{x^{ - 6}} + 2x\) at \(x = - 1\). Solution
  17. The position of an object at any time t is given by \(s\left( t \right) = 3{t^4} - 40{t^3} + 126{t^2} - 9\).
    1. Determine the velocity of the object at any time t.
    2. Does the object ever stop changing?
    3. When is the object moving to the right and when is the object moving to the left?
    Solution
  18. Determine where the function \(h\left( z \right) = 6 + 40{z^3} - 5{z^4} - 4{z^5}\) is increasing and decreasing. Solution
  19. Determine where the function \(R\left( x \right) = \left( {x + 1} \right){\left( {x - 2} \right)^2}\) is increasing and decreasing. Solution
  20. Determine where, if anywhere, the tangent line to \(f\left( x \right) = {x^3} - 5{x^2} + x\) is parallel to the line \(y = 4x + 23\). Solution