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 3.10 : Implicit Differentiation
For problems 1 – 3 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.
- \(\displaystyle \frac{x}{{{y^3}}} = 1\) Solution
- \({x^2} + {y^3} = 4\) Solution
- \({x^2} + {y^2} = 2\) Solution
For problems 4 – 9 find \(y'\) by implicit differentiation.
- \(2{y^3} + 4{x^2} - y = {x^6}\) Solution
- \(7{y^2} + \sin \left( {3x} \right) = 12 - {y^4}\) Solution
- \({{\bf{e}}^x} - \sin \left( y \right) = x\) Solution
- \(4{x^2}{y^7} - 2x = {x^5} + 4{y^3}\) Solution
- \(\cos \left( {{x^2} + 2y} \right) + x\,{{\bf{e}}^{{y^{\,2}}}} = 1\) Solution
- \(\tan \left( {{x^2}{y^4}} \right) = 3x + {y^2}\) Solution
For problems 10 & 11 find the equation of the tangent line at the given point.
- \({x^4} + {y^2} = 3\) at \(\left( {1,\, - \sqrt 2 } \right)\). Solution
- \({y^2}{{\bf{e}}^{2x}} = 3y + {x^2}\) at \(\left( {0,3} \right)\). Solution
For problems 12 & 13 assume that \(x = x\left( t \right)\), \(y = y\left( t \right)\) and \(z = z\left( t \right)\) then differentiate the given equation with respect to t.