Exam 1

1. Question

   What is the derivative of $f(x)=x^8{e}^{3.4x}$, evaluated at $x=0.7$?
   

   Solution

   Using the product rule for $f(x)=g(x)·h(x)$, where $g(x):=x^8$ and $h(x):=e^{3.4x}$, we obtain
   

   ${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{f\text{'}(x)}& =\hfill & [g(x)·h(x)]\text{'}=g\text{'}(x)·h(x)+g(x)·h\text{'}(x)\hfill \\ \multicolumn{1}{c}{}& =\hfill & 8x^{8-1}·e^{3.4x}+x^8·e^{3.4x}·3.4\hfill \\ \multicolumn{1}{c}{}& =\hfill & e^{3.4x}·(8x^7+3.4x^8)\hfill \\ \multicolumn{1}{c}{}& =\hfill & e^{3.4x}·x^7·(8+3.4x).\hfill \end{array}}$

   
   Evaluated at $x=0.7$, the answer is
   

   ${\displaystyle e^{3.4·0.7}·0.7^7·(8+3.4·0.7)=9.236438.}$

   
   Thus, rounded to two digits we have $f\text{'}(0.7)=9.24$.