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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. \(x = 2\)