1. Home
  2. For business economics
  3. Chapter 5: Optimization
  4. Optimization functions of one variable
  5. Second-order derivative
  6. Exercise 2

Exercise 2

$y(x)=5^{3x^2+2}$. Determine $y''(x)$.
$y''(x)=6\cdot 5^{3x^2+2}\cdot \textrm{ln}(5)\cdot (1+\textrm{ln}(5) \cdot 6 x^2)$
$y''(x)=6\cdot 5^{3x^2+2}\cdot (1+\cdot 6 x^2)$
$y''(x)=5^{3x^2+2}\cdot (\textrm{ln}(5))^2$
$y''(x)=6\cdot 5^{3x^2+2}\cdot \textrm{ln}(5)\cdot (1+\textrm{ln}(5) \cdot  x)$
$y(x)=5^{3x^2+2}$. Determine $y''(x)$.
Antwoord 1 correct
Correct
Antwoord 2 optie
$y''(x)=6\cdot 5^{3x^2+2}\cdot (1+\cdot 6 x^2)$
Antwoord 2 correct
Fout
Antwoord 3 optie
$y''(x)=5^{3x^2+2}\cdot (\textrm{ln}(5))^2$
Antwoord 3 correct
Fout
Antwoord 4 optie
$y''(x)=6\cdot 5^{3x^2+2}\cdot \textrm{ln}(5)\cdot (1+\textrm{ln}(5) \cdot  x)$
Antwoord 4 correct
Fout
Antwoord 1 optie
$y''(x)=6\cdot 5^{3x^2+2}\cdot \textrm{ln}(5)\cdot (1+\textrm{ln}(5) \cdot 6 x^2)$
Antwoord 1 feedback
Correct: $y'(x)=5^{3x^2+2}\cdot \textrm{ln}(5)\cdot 6x$, and
$$\begin{align*}
y''(x) & =5^{3x^2+2}\cdot (\textrm{ln}(5))^2\cdot 6x \cdot 6x+5^{3x^2+2}\cdot \textrm{ln}(5)\cdot 6\\
& =6\cdot 5^{3x^2+2}\cdot \textrm{ln}(5)\cdot (1+\textrm{ln}(5) \cdot 6x^2).
\end{align*}$$

Go on.
Antwoord 2 feedback
Wrong: $y'(x)\neq 5^{3x^2+2}\cdot 6x$.

See Derivatives of elementary functions.
Antwoord 3 feedback
Wrong: $y'(x) \neq 5^{3x^2+2}\cdot \textrm{ln}(5)$.

See Chain rule.
Antwoord 4 feedback
Wrong: Do not forget the chain rule when differentiating the first-order derivative.

Try again.