Question

Let $g(x)=\begin{vmatrix}x^{4}&-\cos x&-\sin 2x\\ -8& 4& 3\\ 2& 4& 8\end{vmatrix}$. Then $g'''(0)$ is equal to

• A. -310
• B. 320
• C. -360
• D. -320
• E. -380

Hint:

Math Tip: When rows 2 and 3 of a determinant are purely constants, taking the derivative of the determinant is identical to simply taking the derivative of the terms in row 1 while keeping the constants untouched!

Answer 1

The Correct Option is D

Concept:
Calculus and Matrices - Differentiation of Determinants.
If a determinant has only one row containing variables and the other rows contain constants, you can evaluate the determinant first and then differentiate, or differentiate the variable row directly.
Step 1: Expand the determinant along the first row.
Evaluate $g(x)$ by expanding along Row 1 (since Row 2 and Row 3 are constants): $$ g(x) = x^4 \begin{vmatrix} 4 & 3 \\ 4 & 8 \end{vmatrix} - (-\cos x) \begin{vmatrix} -8 & 3 \\ 2 & 8 \end{vmatrix} + (-\sin 2x) \begin{vmatrix} -8 & 4 \\ 2 & 4 \end{vmatrix} $$
Step 2: Evaluate the $2 \times 2$ constant determinants.

• $D_1 = (4)(8) - (4)(3) = 32 - 12 = 20$
• $D_2 = (-8)(8) - (2)(3) = -64 - 6 = -70$
• $D_3 = (-8)(4) - (2)(4) = -32 - 8 = -40$

Step 3: Construct the simplified function $g(x)$.
Substitute the $2 \times 2$ determinant values back into the expanded equation: $$ g(x) = 20x^4 + \cos x (-70) - \sin 2x (-40) $$ $$ g(x) = 20x^4 - 70\cos x + 40\sin 2x $$
Step 4: Find the first and second derivatives.
Differentiate with respect to $x$ using basic rules: $$ g'(x) = 80x^3 - 70(-\sin x) + 40(2\cos 2x) $$ $$ g'(x) = 80x^3 + 70\sin x + 80\cos 2x $$ Differentiate again: $$ g''(x) = 240x^2 + 70\cos x + 80(-2\sin 2x) $$ $$ g''(x) = 240x^2 + 70\cos x - 160\sin 2x $$
Step 5: Find the third derivative $g'''(x)$.
Differentiate for the third time: $$ g'''(x) = 480x + 70(-\sin x) - 160(2\cos 2x) $$ $$ g'''(x) = 480x - 70\sin x - 320\cos 2x $$
Step 6: Evaluate $g'''(0)$.
Substitute $x = 0$ into the third derivative equation: $$ g'''(0) = 480(0) - 70\sin(0) - 320\cos(0) $$ Since $\sin(0) = 0$ and $\cos(0) = 1$: $$ g'''(0) = 0 - 0 - 320(1) = -320 $$