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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.