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Section 3.6 : Derivatives of Exponential and Logarithm Functions

For problems 1 – 6 differentiate the given function.

1. \(f\left( x \right) = 2{{\bf{e}}^x} - {8^x}\) Solution
2. \(g\left( t \right) = 4{\log _3}\left( t \right) - \ln \left( t \right)\) Solution
3. \(R\left( w \right) = {3^w}\log \left( w \right)\) Solution
4. \(y = {z^5} - {{\bf{e}}^z}\ln \left( z \right)\) Solution
5. \(\displaystyle h\left( y \right) = \frac{y}{{1 - {{\bf{e}}^y}}}\) Solution
6. \(\displaystyle f\left( t \right) = \frac{{1 + 5t}}{{\ln \left( t \right)}}\) Solution
7. Find the tangent line to \(f\left( x \right) = {7^x} + 4{{\bf{e}}^x}\) at \(x = 0\). Solution
8. Find the tangent line to \(f\left( x \right) = \ln \left( x \right){\log _2}\left( x \right)\) at \(x = 2\). Solution
9. Determine if \(\displaystyle V\left( t \right) = \frac{t}{{{{\bf{e}}^t}}}\) is increasing or decreasing at the following points.
   1. \(t = - 4\)
   2. \(t = 0\)
   3. \(t = 10\)
   Solution
10. Determine if \(G\left( z \right) = \left( {z - 6} \right)\ln \left( z \right)\) is increasing or decreasing at the following points.
    1. \(z = 1\)
    2. \(z = 5\)
    3. \(z = 20\)
    Solution