Consider the function $c(q) = 20\sqrt{q} + 5q + 100$. Determine the derivative in $q=4$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$c'(q) = \dfrac{10}{\sqrt{q}} + 5$.
Antwoord 2 correct
Fout
Antwoord 3 optie
This exercise cannot be solved, because the derivative of $\sqrt{q}$ cannot be determined.
Antwoord 3 correct
Fout
Antwoord 4 optie
This exercise cannot be solved, because $c(q)$ is the sum of three functions.
Antwoord 4 correct
Fout
Antwoord 1 optie
$c'(4) = 10$.
Antwoord 1 feedback
Correct: Write $c(q) = u(q) + v(q)$ with $u(q) = 20\sqrt{q} = 20q^{\tfrac{1}{2}}$ and $v(q) = 5q + 100$. Then $v(q)$ is again the sum of two functions: $v(q) = w(q) + y(q)$, with $w(q) = 5q$ and $y(q) = 100$. By the use of the Derivatives of elementary functions and the Scalar product rule we then find $c'(q)$:
$$\begin{align*}
u'(q) &= 20 \cdot \tfrac{1}{2} q^{\tfrac{1}{2}-1} = 10q^{-\tfrac{1}{2}} = \dfrac{10}{\sqrt{q}}\\
w'(q) &= 5\\
y'(q) &= 0\\
v'(q) &= w'(q) + y'(q) = 5\\
c'(q) &= u'(q) + v'(q) = \dfrac{10}{\sqrt{q}} + 5.
\end{align*}$$
Finally, we plug in $q=4$:
$$ c'(4) = \dfrac{10}{\sqrt{4}} + 5= \dfrac{10}{2} + 5 = 5+ 5 = 10.$$
Go on.
$$\begin{align*}
u'(q) &= 20 \cdot \tfrac{1}{2} q^{\tfrac{1}{2}-1} = 10q^{-\tfrac{1}{2}} = \dfrac{10}{\sqrt{q}}\\
w'(q) &= 5\\
y'(q) &= 0\\
v'(q) &= w'(q) + y'(q) = 5\\
c'(q) &= u'(q) + v'(q) = \dfrac{10}{\sqrt{q}} + 5.
\end{align*}$$
Finally, we plug in $q=4$:
$$ c'(4) = \dfrac{10}{\sqrt{4}} + 5= \dfrac{10}{2} + 5 = 5+ 5 = 10.$$
Go on.
Antwoord 2 feedback
Wrong: You have to calculate the derivative in the point $q=4$, not the derivative function $c'(q)$.
Try again.
Try again.
Antwoord 3 feedback
Wrong: The derivative of $\sqrt{q}$ can be determined.
See Properties power functions or Derivatives of elementary functions: Example 2.
See Properties power functions or Derivatives of elementary functions: Example 2.
Antwoord 4 feedback
Wrong: Write $c(q) = u(q) + v(q)$, with $u(q)=20\sqrt{q}$. Then write $v(q) = w(q) + y(q)$ and apply the sum rule twice (once to determine $v'(q)$ and once to determine $c'(q)$).
Try again.
Try again.