Determine the derivative of $y(x) = 2^{5x^2 + 3}$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$y'(x) = \ln(2)2^{5x^2+3}$.
Antwoord 2 correct
Fout
Antwoord 3 optie
$y'(x) = 10x2^{5x^2+3}$.
Antwoord 3 correct
Fout
Antwoord 4 optie
This derivative cannot be determined.
Antwoord 4 correct
Fout
Antwoord 1 optie
$y'(x) = 10\ln(2)x2^{5x^2+3}$.
Antwoord 1 feedback
Correct: In Composite function: Exercise 1 it is shown that $y(x)$ can be written as $u(v(x))$ with $v(x) = 5x^2 + 3$ and $u(v) = 2^v$. By the use of the chain rule we find:
$$
\begin{align*}
v'(x) &= 5\cdot 2x + 0 = 10x\\
u'(v) &= 2^v \ln(2) = \ln(2)2^v\\
y'(x) &= u'\big(v(x)\big)\cdot v'(x) = \ln(2)2^{v(x)} \cdot 10x = 10\ln(2)x2^{5x^2+3}.
\end{align*}
$$
Go on.
$$
\begin{align*}
v'(x) &= 5\cdot 2x + 0 = 10x\\
u'(v) &= 2^v \ln(2) = \ln(2)2^v\\
y'(x) &= u'\big(v(x)\big)\cdot v'(x) = \ln(2)2^{v(x)} \cdot 10x = 10\ln(2)x2^{5x^2+3}.
\end{align*}
$$
Go on.
Antwoord 2 feedback
Antwoord 3 feedback
Wrong: You probably miscalculated $u(v) = 2^v$: $u'(v) \neq 2^v$.
See Derivatives elemenatary functions.
See Derivatives elemenatary functions.
Antwoord 4 feedback
Wrong: The derivative of $y(x)$ can be determined by the use of the chain rule.
See Chain rule, Example 1 and Example 2.
See Chain rule, Example 1 and Example 2.