Differentiation rules (product, quotient, chain) Knowledge-Based Practice

Questions

1. 1.

   If $f(x) = \sqrt{\frac{3x + 1}{2x - 1}}$, what is the value of $f'(1)$?

   • A) $-\frac{5}{2}$
   • B) $\frac{5}{4}$
   • C) $-\frac{5}{4}$
   • D) $-\frac{1}{4}$
2. 2.

   If $f(x) = \sqrt{x + \sqrt{x}}$, what is the value of $f'(4)$?

   • A) $\frac{\sqrt{6}}{12}$
   • B) $\frac{\sqrt{6}}{8}$
   • C) $\frac{5\sqrt{6}}{24}$
   • D) $\frac{5\sqrt{6}}{48}$
3. 3.

   Let $h(x) = \frac{f(2x)}{g(x)}$. If $f(2) = 3$, $f'(2) = -1$, $g(1) = 2$, and $g'(1) = 4$, what is the value of $h'(1)$?

   • A) $-8$
   • B) $-4$
   • C) $2$
   • D) $4$
4. 4.

   If $f(x) = (2x - 1)^3 \sqrt{x^2 + 5}$, what is the value of $f'(2)$?

   • A) 36
   • B) 144
   • C) 180
   • D) 198
5. 5.

   Given the function $f(x) = \frac{(2x + 1)^3}{x^2 + 2}$, what is the value of $f'(1)$?

   • A) 12
   • B) 24
   • C) 3
   • D) 36

Answers and step-by-step solutions appear after you finish the test.