Differentiation rules (product, quotient, chain) Knowledge-Based Download PDF

Questions

1. 1.

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

   • A) $-\frac{7}{8}$
   • B) $\frac{5}{8}$
   • C) $-\frac{5}{8}$
   • D) $\frac{11}{8}$
2. 2.

   Let $f$ and $g$ be differentiable functions such that $f(2) = 3$, $f'(2) = -1$, $g(1) = 2$, and $g'(1) = 4$. If $h(x) = x^3 f(g(x))$, what is the value of $h'(1)$?

   • A) -4
   • B) 8
   • C) 5
   • D) 13
3. 3.

   A differentiable function $f$ satisfies $f(7) = 10$ and $f'(7) = 2$. If $h(x) = \frac{f(x^3 - 1)}{2x + 1}$, what is the value of $h'(2)$?

   • A) $-4$
   • B) $-\frac{2}{5}$
   • C) $\frac{28}{5}$
   • D) $4$
4. 4.

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

   • A) 40
   • B) 60
   • C) 144
   • D) 72
5. 5.

   Given the function $f(x) = \frac{(3x - 1)^4}{x^2 + 1}$, what is the value of $f'(1)$?

   • A) 8
   • B) 48
   • C) 56
   • D) 40