Computing a derivative (bonus)

Optional bonus: differentiation rules, partial derivatives and the exact slope of the crêpe and admission errors

12 exercises

Each exercise is graded automatically, in your browser. The full statements are available once you have an account.

1. Lesson: slope of a formula

   In level 2, your zoom machine found the best crêpe line by trying finer and finer settings. With two settings, it is fast. With ten thousand, trying is no longer enough: you need to know which way to move each…

2. Exact slope

   The on-board calculator of the crêpe stand tracks five curves, all functions of the same setting $x$: - sales according to temperature: $f_1(x) = 2x^3 - 5x^2 + x - 7$; - attendance after opening: $f_2(x) = x\,e^{-x}$; -…

3. Slope of the squasher

   In level 2, the admissions table passed each score $z$ through the function you invented to squash any number between 0 and 1: $\sigma(z) = \dfrac{1}{1 + e^{-z}}$. The table's sensor only transmits the output…

4. Wrong step

   A student differentiates $f(x) = x^2 e^{-x}$: 1. $f = u \cdot v$ with $u = x^2$ and $v = e^{-x}$; 2. $u' = 2x$; 3. $v' = e^{-x}$; 4. $f'(x) = u'v + uv' = 2x\,e^{-x} + x^2 e^{-x}$. Which step is wrong?

5. Second wrong step

   A student differentiates $f(x) = \ln(1 + x^2)$, then computes the slope at $x = 1$: 1. $f = \ln u$ with $u = 1 + x^2$; 2. $(\ln u)' = \frac{1}{u}$; 3. so $f'(x) = \frac{1}{1 + x^2}$; 4. $f'(1) = \frac{1}{2}$. Which step…

6. Point on the line

   To tune the line of the lying points, we compute the slope of the mean absolute error with respect to the setting. For each point, the slope of $|r|$ (where $r$ is the gap between the point and the line) is $+1$ if…

7. Lesson: several settings

   The error of the crêpe line depends on two settings, $a$ and $b$. To know how it reacts to $a$ alone, we freeze $b$: it becomes a constant, and we differentiate with respect to $a$ with the usual rules. The…

8. Two settings

   The stand's dashboard shows three indicators that depend on two settings $a$ and $b$: - $f_1(a, b) = a^3 b - 2ab^2 + 5b$; - $f_2(a, b) = e^{ab} - a$; - $f_3(a, b) = (14a + b - 190)^2$, the squared gap of the line…

9. Slope of the error

   Go back to the crêpe line $y = ax + b$ and to the lying points. To tune the line, the calculator needs the exact slope of the error with respect to each of the two settings, over $n$ records $(x_i, y_i)$: -…

10. Slope of admission

    The school's admission board predicts whether a student is admitted from their grade $x$ with $p = \sigma(w x + b)$, where $\sigma(z) = \frac{1}{1 + e^{-z}}$. Over $n$ files $(x_i, y_i)$, where $y_i$ is 1 if the student…

11. Lesson: what you can differentiate

    You have differentiated whole losses. With the gap $r_i = \hat{y}_i - y_i$ of each record: - the slope of the mean of the squares of a line is $\frac{2}{n}\sum_i r_i\,(x_i, 1)$; - that of the mean of the absolute gaps…

12. Two thousand sensors

    The crêpe line has grown: the calculator now predicts $\hat{y} = Xw$ from $d$ sensors, over $n$ records. $X$ has $n$ rows (the records) and $d$ columns (the sensors), $w$ holds the $d$ settings, and the error is the…