Show All Notes Hide All Notes

Section 3.10 : Implicit Differentiation

For problems 1 – 6 do each of the following.

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. \(\cos \left( {{z^2}{x^3}} \right) + \sqrt {{y^2} + {x^2}} = 0\)