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Section 3.13 : Logarithmic Differentiation

For problems 1 – 6 use logarithmic differentiation to find the first derivative of the given function.

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

For problems 7 – 10 find the first derivative of the given function.

1. \(y = {x^{\,\ln \left( x \right)}}\)
2. \(R\left( t \right) = {\left[ {\sin \left( {4t} \right)} \right]^{6\,t}}\)
3. \(h\left( w \right) = {\left( {6 - {w^2}} \right)^{2 + 8w + {w^{\,3}}}}\)
4. \(g\left( z \right) = {z^2}{\left[ {3 + z} \right]^{1 - {z^{\,2}}}}\)