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Section 3.10 : Implicit Differentiation

For problems 1 – 3 do each of the following.

1. \(\displaystyle \frac{x}{{{y^3}}} = 1\) Solution
2. \({x^2} + {y^3} = 4\) Solution
3. \({x^2} + {y^2} = 2\) Solution

For problems 4 – 9 find \(y'\) by implicit differentiation.

1. \(2{y^3} + 4{x^2} - y = {x^6}\) Solution
2. \(7{y^2} + \sin \left( {3x} \right) = 12 - {y^4}\) Solution
3. \({{\bf{e}}^x} - \sin \left( y \right) = x\) Solution
4. \(4{x^2}{y^7} - 2x = {x^5} + 4{y^3}\) Solution
5. \(\cos \left( {{x^2} + 2y} \right) + x\,{{\bf{e}}^{{y^{\,2}}}} = 1\) Solution
6. \(\tan \left( {{x^2}{y^4}} \right) = 3x + {y^2}\) Solution

For problems 10 & 11 find the equation of the tangent line at the given point.

1. \({x^4} + {y^2} = 3\) at \(\left( {1,\, - \sqrt 2 } \right)\). Solution
2. \({y^2}{{\bf{e}}^{2x}} = 3y + {x^2}\) 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.

1. \({x^2} - {y^3} + {z^4} = 1\) Solution
2. \({x^2}\cos \left( y \right) = \sin \left( {{y^3} + 4z} \right)\) Solution