I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 3.10 : Implicit Differentiation
For problems 1 – 6 do each of the following.
- Find \(y'\) by solving the equation for y and differentiating directly.
- Find \(y'\) by implicit differentiation.
- Check that the derivatives in (a) and (b) are the same.
- \({x^2}{y^9} = 2\)
- \(\displaystyle \frac{{6x}}{{{y^7}}} = 4\)
- \(1 = {x^4} + 5{y^3}\)
- \(8x - {y^2} = 3\)
- \(4x - 6{y^2} = x{y^2}\)
- \(\ln \left( {x\,y} \right) = x\)
For problems 7 – 21 find \(y'\) by implicit differentiation.
- \({y^2} - 12{x^3} = 8y\)
- \(3{y^7} + {x^{10}} = {y^{ - 2}} - 6{x^3} + 2\)
- \({y^{ - 3}} + 4{x^{ - 1}} = 8{y^{ - 1}}\)
- \(10{x^4} - {y^{ - 6}} = 7{y^3} + 4{x^{ - 3}}\)
- \(\sin \left( x \right) + \cos \left( y \right) = {{\bf{e}}^{4y}}\)
- \(x + \ln \left( y \right) = \sec \left( y \right)\)
- \({y^2}\left( {4 - {x^2}} \right) = {y^7} + 9x\)
- \(6{x^{ - 2}} - {x^3}{y^2} + 4x = 0\)
- \(8xy + 2{x^4}{y^{ - 3}} = {x^3}\)
- \(y x^3 - \cos \left( x \right)\sin \left( y \right) = 7x\)
- \({{\bf{e}}^x}\cos \left( y \right) + \sin \left( {xy} \right) = 9\)
- \({x^2} + \sqrt {{x^3} + 2y} = {y^2}\)
- \(\tan \left( {3x + 7y} \right) = 6 - 4{x^{ - 1}}\)
- \({{\bf{e}}^{{x^{\,2}} + {y^{\,2}}}} = {{\bf{e}}^{{x^{\,2}}{y^{\,2}}}} + 1\)
- \(\displaystyle \sin \left( {\frac{x}{y}} \right) + {x^3} = 2 - {y^4}\)
For problems 22 - 24 find the equation of the tangent line at the given point.
- \(3x + {y^2} = {x^2} - 19\) at \(\left( { - 4,3} \right)\)
- \({x^2}y = {y^2} - 6x\) at \(\left( {2,6} \right)\)
- \(2\sin \left( x \right)\cos \left( y \right) = 1\) 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.
- \({x^2} - {y^3} = 4y + 9\) at \(\left( {2, - 1} \right)\)
- \({{\bf{e}}^{1 - x}}{{\bf{e}}^{{y^{\,2}}}} = {x^3} + y\) at \(\left( {1,0} \right)\)
- \(\sin \left( {\pi - x} \right) + {y^2}\cos \left( x \right) = y\) 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.
- \({x^4} - 6z = 3 - {y^2}\)
- \(x\,{y^4} = {y^2}{z^3}\)
- \({z^7}{{\bf{e}}^{6\,y}} = {\left( {{y^2} - 8x} \right)^{10}} + {z^{ - 4}}\)
- \(\cos \left( {{z^2}{x^3}} \right) + \sqrt {{y^2} + {x^2}} = 0\)