sc <- NULL while(is.null(sc)) { ## parameters a <- sample(2:9, 1) b <- sample(seq(2, 4, 0.1), 1) c <- sample(seq(0.6, 0.9, 0.01), 1) ## solution res <- exp(b * c) * (a * c^(a-1) + b * c^a) ## schoice err <- c(a * c^(a-1) * exp(b * c), a * c^(a-1) * exp(b * c) + c^a * exp(b * c)) rg <- if(res < 4) c(0.5, 5.5) else res * c(0.5, 1.5) sc <- num_to_schoice(res, wrong = err, range = rg, delta = 0.1) }
What is the derivative of $f(x) = x^{r a
} e^{r b
x}$, evaluated at $x = r c
$?
answerlist(sc$questions, markup = "markdown")
Using the product rule for $f(x) = g(x) \cdot h(x)$, where $g(x) := x^{r a
}$ and $h(x) := e^{r b
x}$, we obtain
$$
\begin{aligned}
f'(x) &= [g(x) \cdot h(x)]' = g'(x) \cdot h(x) + g(x) \cdot h'(x) \
&= r a
x^{r a
- 1} \cdot e^{r b
x} + x^{r a
} \cdot e^{r b
x} \cdot r b
\
&= e^{r b
x} \cdot(r a
x^r a-1
+ r b
x^{r a
}) \
&= e^{r b
x} \cdot x^r a-1
\cdot (r a
+ r b
x).
\end{aligned}
$$
Evaluated at $x = r c
$, the answer is
$$ e^{r b
\cdot r c
} \cdot r c
^r a-1
\cdot (r a
+ r b
\cdot r c
) = r fmt(res, 6)
. $$
Thus, rounded to two digits we have $f'(r c
) = r fmt(res)
$.
answerlist(ifelse(sc$solutions, "True", "False"), markup = "markdown")
extype: schoice
exsolution: r mchoice2string(sc$solutions)
exname: derivative exp
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