Mathematics for Management -- Supplementary Electronic Materials

Quiz: The Chain Rule

After having worked through the electronic materials (set of notes and video clips), please self-evaluate your learning progress by completing the following quiz. Therefore, please check the correct answer of each of the given questions (single choice) and press the submit button after you completed all 10 questions. (Sometimes the page needs a reload such that the grading script starts to work.)

Your selections will persist so that you can review your answers in case you did not achieve the complete grade. At the bottom of the page you can also find the solution key.

1. Determine the derivative \( f'(x)\) of \( f(x) = \left( 5x^2 - 2x \right)^3\).

\( f'(x) = 3 \left( 10x - 2 \right)^2 \) \( f'(x) = 10 \left( 5x^2 - 2x \right)^2 \) \( f'(x) = \left( 30x - 6 \right) \left( 5x^2 - 2x \right)^2 \) \( f'(x) = \left( 10x - 2 \right) \left( 5x^2 - 2x \right)^2 \)

2. Let \(f(x)\) and \(g(x)\) be continuously differentiable functions such that \(f(2) = 3\), \(f'(2) = 7\), \(f'(3) = 13\), \(f'(5) = 19\), \(g(2) = 5\), \(g'(2) = 11\), \(g'(3) = 17\), and \(g'(5) = 23\). Let \(F(x) = g(f(x))\), then what is the value of \(F'(2)\)?

\( 161 \) \( 143 \) \( 209 \) \( 119 \)

3. Determine the derivative \(f'(x)\) of \(f(x) = \frac{x}{\sqrt{1-x^2}}\).

\( f'(x) = \frac{1-2x^2}{(1-x^2)^{3/2}} \) \( f'(x) = \frac{1}{1-x^2} \) \( f'(x) = \frac{1-2x^2}{(1-x^2)^{1/2}} \) \( f'(x) = \frac{1}{(1-x^2)^{3/2}} \)

4. Let \(f(x) = 4x {\rm{e}}^{-x^2}\). Determine the value of \(f''(1)\).

\( 4 {\rm{e}}^{-1} \) \( -4 {\rm{e}} \) \( -8 \) \( -8 {\rm{e}}^{-1} \)

5. Determine the derivative \(f'(x)\) of \(f(x) = \frac{1}{2 \sin(2x)}\).

\( f'(x) = -\csc(2x) \cot(2x) \) \( f'(x) = -\csc^2(2x) \) \( f'(x) = 4 \csc(2x) \cot(2x) \) \( f'(x) = \frac{\cos(2x)}{2 \sqrt{\sin(2x)}} \)

6. Determine the derivative \(f'(x)\) of \(f(x) = \sin(11x) \cdot \cos(11x)\).

\( f'(x) = 11 \cos(11x) \) \( f'(x) = -121 \sin^2(11x) \) \( f'(x) = 11 \cos(22x) \) \( f'(x) = 11 \cos^2(11x) - 11 \sin^2(11x) \)

7. Determine the second derivative \(f''(x)\) of \(f(x) = \sin(x) \cdot {\rm{e}}^{2x}\).

\( f''(x) = - 4 \sin(x) {\rm{e}}^{2x} \) \( f''(x) = -\sin(x) {\rm{e}}^{2x} + 2 \cos(x) {\rm{e}}^{2x} \) \( f''(x) = -\sin(x) {\rm{e}}^{2x} + 2 \cos(x) {\rm{e}}^{2x} + 4 \sin(x) {\rm{e}}^{2x} \) \( f''(x) = -\sin(x) {\rm{e}}^{2x} + 4 \cos(x) {\rm{e}}^{2x} + 4 \sin(x) {\rm{e}}^{2x} \)

8. Let \(f(x) = \frac{1}{x^2 + 1}\) and \(g(x) = \sqrt{x}\), then the derivative \(F'(x)\) of \(F(x) = f(g(x))\) is

\( F'(x) = \frac{-\sqrt{x}}{(x^2 + 1)^2} \) \( F'(x) = -(x+1)^{-2} \) \( F'(x) = \frac{-2x}{(x^2+1)} \) \( F'(x) = \frac{1}{2 \sqrt{x} (x+1)} \)

9. Let \(f(u) = \sin(u)\) and \(u = g(x) = x^2 - 9\), then what value takes \((f \circ g)'(3)\)?

\( 0 \) \( 1 \) \( 6 \) \( 9 \)

10. Determine the second derivative \(f''(x)\) of \(f(x) = a \sin(cx) + b \cos(cx)\), where \(a, b \in \mathbb{R}\) are constants.

\( f''(x) = ac^2 \sin(cx) + bc^2 \cos(cx) \) \( f''(x) = a^2 c^2 \sin(cx) - b^2 c^2 \cos(cx) \) \( f''(x) = -a^2 c^2 \sin(cx) + b^2 c^2 \cos(cx) \) \( f''(x) = -ac^2 \sin(cx) - bc^2 \cos(cx) \)

Your grade is: __ of 10

In case you cannot find the correct answers even after reviewing the electronic materials again, please click on the "+" sign to read more.

Solution: 1c; 2c; 3d; 4d; 5a; 6d; 7d; 8b; 9c; 10d

Here, a, b, c, d indicate the 1st, 2nd, 3rd, and 4th answer choice, respectively, for the numbered questions.