Quiz 3, MAT229

Make sure that your name is on your paper. There are 10 points total. Show work!

1. (3 pts) Compute the derivative of \(f(x)\) with respect to \(x\): \[ f \left(\right. x \left.\right) = \text{ln} \sqrt{5 x - 7} \]
   
   
   
   
   
   
   
   
2. (4 pts) Find an equation of the tangent line of to the graph of \(f\) at \(x=-1\), where \[ f \left(\right. x \left.\right) = 4 x e^{\left(\right. x^{2} - 1 \left.\right)} \]
   
   
   
   
   
   
   
   
3. (3 pts) Every rational function behaves like a polynomial far from the origin. What particular polynomial does this rational function behave like as \(x \to \infty\)?

   \[ \frac{3x^3-x^2+4x-3}{x-1} \]

   
   
   
   
   
   
   
   
   

4. Extra credit (1 pt): find the derivative of the log function \(\ln(x)\) by using the formula for the derivative of the inverse function of \(e^x.\)