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### Section 3-6 : Derivatives of Exponential and Logarithm Functions

For problems 1 – 12 differentiate the given function.

1. $$g\left( z \right) = {10^z} - {9^z}$$
2. $$f\left( x \right) = 9{\log _4}\left( x \right) + 12{\log _{11}}\left( x \right)$$
3. $$h\left( t \right) = {6^t} - 4{{\bf{e}}^t}$$
4. $$R\left( x \right) = 20\ln \left( x \right) + {\log _{123}}\left( x \right)$$
5. $$Q\left( t \right) = \left( {{t^2} - 6t + 3} \right){{\bf{e}}^t}$$
6. $$y = v + {8^v}\,{9^v}$$
7. $$U\left( z \right) = {\log _4}\left( z \right) - {z^6}\ln \left( z \right)$$
8. $$h\left( x \right) = {\log _3}\left( x \right)\log \left( x \right)$$
9. $$\displaystyle f\left( w \right) = \frac{{1 - {{\bf{e}}^w}}}{{1 + 7{{\bf{e}}^w}}}$$
10. $$\displaystyle f\left( t \right) = \frac{{1 + 4\ln \left( t \right)}}{{5{t^3}}}$$
11. $$\displaystyle g\left( r \right) = \frac{{{r^2} + {{\log }_7}\left( r \right)}}{{{7^r}}}$$
12. $$\displaystyle V\left( t \right) = \frac{{{t^4}{{\bf{e}}^t}}}{{\ln \left( t \right)}}$$
13. Find the tangent line to $$f\left( x \right) = \left( {1 - 8x} \right){{\bf{e}}^x}$$ at $$x = - 1$$.
14. Find the tangent line to $$f\left( x \right) = 3{x^2}\ln \left( x \right)$$ at $$x = 1$$.
15. Find the tangent line to $$f\left( x \right) = 3{{\bf{e}}^x} + 8\ln \left( x \right)$$ at $$x = 2$$.
16. Determine if $$U\left( y \right) = {4^y} - 3{{\bf{e}}^y}$$ is increasing or decreasing at the following points.
1. $$y = - 2$$
2. $$y = 0$$
3. $$y = 3$$
17. Determine if $$\displaystyle y\left( z \right) = \frac{{{z^2}}}{{\ln \left( z \right)}}$$ is increasing or decreasing at the following points.
1. $$z = \frac{1}{2}$$
2. $$z = 2$$
3. $$z = 6$$
18. Determine if $$h\left( x \right) = {x^2}{{\bf{e}}^x}$$ is increasing or decreasing at the following points.
1. $$x = -1$$
2. $$x = 0$$
3. $$x = 2$$