Question

Let \( f_1(x), f_2(x), g_1(x), g_2(x) \) be differentiable functions on \( \mathbb{R} \). Let 

Then \( F'(x) \) is equal to 
 

• A. \( f_1'(x) f_2'(x) + f_1(x) g_1'(x) \)
• B. \( f_1'(x) f_2'(x) + f_1(x) g_1'(x) \)
• C. \( f_1'(x) f_2'(x) - |f_1(x) g_1'(x)| \)
• D. \( f_1'(x) g_2'(x) - f_2'(x) g_2'(x) \)

Hint:

When differentiating a 2x2 matrix determinant, apply the rule for matrix determinants and differentiate each element with respect to \( x \).

Answer 1

The Correct Option is B

Step 1: Understanding the determinant. 
We are given a determinant of a 2x2 matrix:

The derivative of this determinant with respect to \( x \) involves the derivatives of \( f_1(x), f_2(x), g_1(x), g_2(x) \). Using the rule for differentiating a 2x2 determinant, we get: \[ F'(x) = \frac{d}{dx} \left[ f_1(x)g_2(x) - f_2(x)g_1(x) \right]. \] 
Step 2: Apply the derivative rule. 
Differentiating each term gives: \[ F'(x) = f_1'(x) g_2(x) + f_1(x) g_2'(x) - f_2'(x) g_1(x) - f_2(x) g_1'(x). \] This matches option (B). 

Step 3: Conclusion. 
Thus, the correct answer is (B).