Pauls Online Notes
Pauls Online Notes
Home / Calculus I / Derivatives / Implicit Differentiation
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.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

Section 3-10 : Implicit Differentiation

For problems 1 – 6 do each of the following.

  1. Find \(y'\) by solving the equation for y and differentiating directly.
  2. Find \(y'\) by implicit differentiation.
  3. Check that the derivatives in (a) and (b) are the same.

  1. \({x^2}{y^9} = 2\)
  2. \(\displaystyle \frac{{6x}}{{{y^7}}} = 4\)
  3. \(1 = {x^4} + 5{y^3}\)
  4. \(8x - {y^2} = 3\)
  5. \(4x - 6{y^2} = x{y^2}\)
  6. \(\ln \left( {x\,y} \right) = x\)

For problems 7 – 21 find \(y'\) by implicit differentiation.

  1. \({y^2} - 12{x^3} = 8y\)
  2. \(3{y^7} + {x^{10}} = {y^{ - 2}} - 6{x^3} + 2\)
  3. \({y^{ - 3}} + 4{x^{ - 1}} = 8{y^{ - 1}}\)
  4. \(10{x^4} - {y^{ - 6}} = 7{y^3} + 4{x^{ - 3}}\)
  5. \(\sin \left( x \right) + \cos \left( y \right) = {{\bf{e}}^{4y}}\)
  6. \(x + \ln \left( y \right) = \sec \left( y \right)\)
  7. \({y^2}\left( {4 - {x^2}} \right) = {y^7} + 9x\)
  8. \(6{x^{ - 2}} - {x^3}{y^2} + 4x = 0\)
  9. \(8xy + 2{x^4}{y^{ - 3}} = {x^3}\)
  10. \(y x^3 - \cos \left( x \right)\sin \left( y \right) = 7x\)
  11. \({{\bf{e}}^x}\cos \left( y \right) + \sin \left( {xy} \right) = 9\)
  12. \({x^2} + \sqrt {{x^3} + 2y} = {y^2}\)
  13. \(\tan \left( {3x + 7y} \right) = 6 - 4{x^{ - 1}}\)
  14. \({{\bf{e}}^{{x^{\,2}} + {y^{\,2}}}} = {{\bf{e}}^{{x^{\,2}}{y^{\,2}}}} + 1\)
  15. \(\displaystyle \sin \left( {\frac{x}{y}} \right) + {x^3} = 2 - {y^4}\)

For problems 22 - 24 find the equation of the tangent line at the given point.

  1. \(3x + {y^2} = {x^2} - 19\) at \(\left( { - 4,3} \right)\)
  2. \({x^2}y = {y^2} - 6x\) at \(\left( {2,6} \right)\)
  3. \(2\sin \left( x \right)\cos \left( y \right) = 1\) at \(\displaystyle \left( {\frac{\pi }{4}, - \frac{\pi }{4}} \right)\)

For problems 25 – 27 determine if the function is increasing, decreasing or not changing at the given point.

  1. \({x^2} - {y^3} = 4y + 9\) at \(\left( {2, - 1} \right)\)
  2. \({{\bf{e}}^{1 - x}}{{\bf{e}}^{{y^{\,2}}}} = {x^3} + y\) at \(\left( {1,0} \right)\)
  3. \(\sin \left( {\pi - x} \right) + {y^2}\cos \left( x \right) = y\) at \(\displaystyle \left( {\frac{\pi }{2},1} \right)\)

For problems 28 - 31 assume that \(x = x\left( t \right)\), \(y = y\left( t \right)\) and \(z = z\left( t \right)\) and differentiate the given equation with respect to t.

  1. \({x^4} - 6z = 3 - {y^2}\)
  2. \(x\,{y^4} = {y^2}{z^3}\)
  3. \({z^7}{{\bf{e}}^{6\,y}} = {\left( {{y^2} - 8x} \right)^{10}} + {z^{ - 4}}\)
  4. \(\cos \left( {{z^2}{x^3}} \right) + \sqrt {{y^2} + {x^2}} = 0\)