Consider the function $y(x)=\sqrt{4x+1}$. Calculate the derivative in $x=2$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$y'(2) = \dfrac{1}{6}$.
Antwoord 2 correct
Fout
Antwoord 3 optie
$y'(2) = \dfrac{1}{4}$.
Antwoord 3 correct
Fout
Antwoord 4 optie
$y'(2)$ cannot be determined, because none of the rules of differentiation can be applied to $y(x)$.
Antwoord 4 correct
Fout
Antwoord 1 optie
$y'(2) = \dfrac{2}{3}$.
Antwoord 1 feedback
Correct: Write $y(x) = \big(u(x)\big)^p$, with $u(x) = 4x+1$ and $p=\tfrac{1}{2}$. Then:
$$
\begin{align*}
u'(x) &= 4\\
y'(x) &= p\big(u(x)\big)^{p-1} \cdot u'(x)\\
&= \dfrac{1}{2} \big(4x+1\big)^{\tfrac{1}{2}-1}\cdot 4 = 2(4x+1)^{-\tfrac{1}{2}} = \dfrac{2}{\sqrt{4x+1}}\\
y'(2) &= \dfrac{2}{\sqrt{4\cdot 2 + 1}} = \dfrac{2}{\sqrt{9}} = \dfrac{2}{3}.
\end{align*}
$$
Go on.
$$
\begin{align*}
u'(x) &= 4\\
y'(x) &= p\big(u(x)\big)^{p-1} \cdot u'(x)\\
&= \dfrac{1}{2} \big(4x+1\big)^{\tfrac{1}{2}-1}\cdot 4 = 2(4x+1)^{-\tfrac{1}{2}} = \dfrac{2}{\sqrt{4x+1}}\\
y'(2) &= \dfrac{2}{\sqrt{4\cdot 2 + 1}} = \dfrac{2}{\sqrt{9}} = \dfrac{2}{3}.
\end{align*}
$$
Go on.
Antwoord 2 feedback
Antwoord 3 feedback
Wrong: You did not apply the composite power in the correct way.
See Extra explanation: special cases.
See Extra explanation: special cases.
Antwoord 4 feedback
Wrong: Write $y(x) = \big(u(x)\big)^p$. Then you can use the composite power rule.
See Extra explanation: special cases and Properties power functions.
See Extra explanation: special cases and Properties power functions.