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Section 3-12 : Higher Order Derivatives

For problems 1 – 9 determine the fourth derivative of the given function.

1. $$f\left( z \right) = {z^8} + 2{z^6} - 7{z^4} + 20{z^2} - 3$$
2. $$y = 6{t^4} - 5{t^3} + 4{t^2} - 3t + 2$$
3. $$V\left( t \right) = 6{t^{ - 2}} + 7{t^{ - 3}} - {t^{ - 4}}$$
4. $$g\left( x \right) = \frac{3}{x} - \frac{1}{{4{x^3}}} + \frac{3}{{2{x^5}}}$$
5. $$h\left( x \right) = 8\sqrt x - \sqrt[3]{x} + 5\,\,\sqrt[4]{{{x^9}}}$$
6. $$\displaystyle h\left( y \right) = \sqrt[3]{{{y^2}}} - \frac{{32}}{{\sqrt[4]{y}}} + \frac{1}{{3\sqrt {{y^5}} }}$$
7. $$\displaystyle y = 9\sin \left( z \right) - \sin \left( {4z} \right) + 7\cos \left( \frac{2x}{3} \right)$$
8. $$R\left( x \right) = 2{{\bf{e}}^{ - x}} - 3{{\bf{e}}^{1 + 8x}} + 9\ln \left( {6x} \right)$$
9. $$f\left( t \right) = \ln \left( {{t^6}} \right) - \cos \left( {4t} \right) + 9\sin \left( {2t} \right) + {{\bf{e}}^{7t}}$$

For problems 10 – 20 determine the second derivative of the given function.

1. $$Q\left( w \right) = \cos \left( {2 - 7{w^2}} \right)$$
2. $$f\left( z \right) = \sin \left( {1 + {{\bf{e}}^{2x}}} \right)$$
3. $$y = \tan \left( {3x} \right)$$
4. $$z = \csc \left( {8w} \right)$$
5. $$f\left( u \right) = {{\bf{e}}^{4{u^{\,2}} + 9u}}$$
6. $$h\left( x \right) = \ln \left( {{x^2} - 3x} \right)$$
7. $$g\left( z \right) = \ln \left( {3 + \cos \left( z \right)} \right)$$
8. $$\displaystyle f\left( x \right) = \frac{1}{{\sqrt {6x + {x^4}} }}$$
9. $$f\left( x \right) = {\left[ {3\sin \left( x \right) + 8\cos \left( {2x} \right)} \right]^{ - 3}}$$
10. $$f\left( t \right) = {\sin ^3}\left( {2t} \right)$$
11. $$A\left( w \right) = {\tan ^4}\left( w \right)$$

For problems 21 – 23 determine the third derivative of the given function.

1. $$g\left( x \right) = \sec \left( {3x} \right)$$
2. $$y = {{\bf{e}}^{1 - 2{t^{\,3}}}}$$
3. $$h\left( w \right) = \cos \left( {w - {w^2}} \right)$$

For problems 24 - 27 determine the second derivative of the given function.

1. $$6y - {y^2} = 3{x^4} + 9x$$
2. $${y^3} - 4{x^2} = 11x - 2{y^2}$$
3. $${{\bf{e}}^y} + 4x = {y^3} - 1$$
4. $$y\cos \left( x \right) = 3 + 4{y^2}$$