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### Section 3.10 : Implicit Differentiation

For problems 1 – 6 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. $${x^2}{y^9} = 2$$
2. $$\displaystyle \frac{{6x}}{{{y^7}}} = 4$$
3. $$1 = {x^4} + 5{y^3}$$
4. $$8x - {y^2} = 3$$
5. $$4x - 6{y^2} = x{y^2}$$
6. $$\ln \left( {x\,y} \right) = x$$

For problems 7 – 21 find $$y'$$ by implicit differentiation.

1. $${y^2} - 12{x^3} = 8y$$
2. $$3{y^7} + {x^{10}} = {y^{ - 2}} - 6{x^3} + 2$$
3. $${y^{ - 3}} + 4{x^{ - 1}} = 8{y^{ - 1}}$$
4. $$10{x^4} - {y^{ - 6}} = 7{y^3} + 4{x^{ - 3}}$$
5. $$\sin \left( x \right) + \cos \left( y \right) = {{\bf{e}}^{4y}}$$
6. $$x + \ln \left( y \right) = \sec \left( y \right)$$
7. $${y^2}\left( {4 - {x^2}} \right) = {y^7} + 9x$$
8. $$6{x^{ - 2}} - {x^3}{y^2} + 4x = 0$$
9. $$8xy + 2{x^4}{y^{ - 3}} = {x^3}$$
10. $$y x^3 - \cos \left( x \right)\sin \left( y \right) = 7x$$
11. $${{\bf{e}}^x}\cos \left( y \right) + \sin \left( {xy} \right) = 9$$
12. $${x^2} + \sqrt {{x^3} + 2y} = {y^2}$$
13. $$\tan \left( {3x + 7y} \right) = 6 - 4{x^{ - 1}}$$
14. $${{\bf{e}}^{{x^{\,2}} + {y^{\,2}}}} = {{\bf{e}}^{{x^{\,2}}{y^{\,2}}}} + 1$$
15. $$\displaystyle \sin \left( {\frac{x}{y}} \right) + {x^3} = 2 - {y^4}$$

For problems 22 - 24 find the equation of the tangent line at the given point.

1. $$3x + {y^2} = {x^2} - 19$$ at $$\left( { - 4,3} \right)$$
2. $${x^2}y = {y^2} - 6x$$ at $$\left( {2,6} \right)$$
3. $$2\sin \left( x \right)\cos \left( y \right) = 1$$ 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.

1. $${x^2} - {y^3} = 4y + 9$$ at $$\left( {2, - 1} \right)$$
2. $${{\bf{e}}^{1 - x}}{{\bf{e}}^{{y^{\,2}}}} = {x^3} + y$$ at $$\left( {1,0} \right)$$
3. $$\sin \left( {\pi - x} \right) + {y^2}\cos \left( x \right) = y$$ 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.

1. $${x^4} - 6z = 3 - {y^2}$$
2. $$x\,{y^4} = {y^2}{z^3}$$
3. $${z^7}{{\bf{e}}^{6\,y}} = {\left( {{y^2} - 8x} \right)^{10}} + {z^{ - 4}}$$
4. $$\cos \left( {{z^2}{x^3}} \right) + \sqrt {{y^2} + {x^2}} = 0$$