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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
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.
- \({x^4} - 6z = 3 - {y^2}\)
- \(x\,{y^4} = {y^2}{z^3}\)
- \({z^7}{{\bf{e}}^{6\,y}} = {\left( {{y^2} - 8x} \right)^{10}} + {z^{ - 4}}\)
- \(\cos \left( {{z^2}{x^3}} \right) + \sqrt {{y^2} + {x^2}} = 0\)