## 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 $$f(x) = g(x) \cdot h(x)$$, where $$g(x) := x^{8}$$ and $$h(x) := e^{3.4x}$$, we obtain \begin{aligned} f'(x) & = & [g(x) \cdot h(x)]' = g'(x) \cdot h(x) + g(x) \cdot h'(x) \\ & = & 8 x^{8 - 1} \cdot e^{3.4x} + x^{8} \cdot e^{3.4x} \cdot 3.4 \\ & = & e^{3.4x} \cdot(8 x^7 + 3.4 x^{8}) \\ & = & e^{3.4x} \cdot x^7 \cdot (8 + 3.4x).\end{aligned} Evaluated at $$x = 0.7$$, the answer is $e^{3.4\cdot 0.7} \cdot 0.7^7 \cdot (8 + 3.4\cdot 0.7) = 9.236438.$ Thus, rounded to two digits we have $$f'(0.7) = 9.24$$.