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### Section 3.10 : Implicit Differentiation

For problems 1 – 3 do each of the following.

1. Find $$y'$$ by solving the equation for y and differentiating directly.
2. Find $$y'$$ by implicit differentiation.
3. Check that the derivatives in (a) and (b) are the same.

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