First slide

Derivatives

Question

$f (x) = |\begin{array}{ccc} x^{2} & 2 x + 1 & 2 x^{3} - 3 \\ \sin x & \cos x & - 1 \\ 2 & 3 & x \end{array}| \underline{If then} the coefficent of x in f^{'} (x) a t x = 0$

Moderate

A

0
B

-1
C

13
D

-13

Solution

$Coefficient of x in f (x) is f^{'} (0)$
$\begin{array}{l} ∴ f^{'} (x) = |\begin{array}{ccc} 2 x & 2 & 6 x^{2} \\ \sin x & \cos x & - 1 \\ 2 & 3 & x \end{array}| + |\begin{array}{ccc} x^{2} & 2 x + 1 & 2 x^{3} - 3 \\ \cos x & - \sin x & 0 \\ 2 & 3 & x \end{array}| + |\begin{array}{ccc} x^{2} & 2 x + 1 & 2 x^{3} - 3 \\ \sin x & \cos x & - 1 \\ 0 & 0 & 1 \end{array}| \\ f^{'} (0) = - 13 \end{array}$

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