Question

If f(x) = |(cosx, x , 1), (2sinx, x², 2x), (tanx, x, 1)|, then lim(x → 0) f'(x)/x

(A) does not exist. 

(B) exists and is equal to 2. 

(C) exists and is equal to 0. 

(D) exists and is equal to −2.

Answer 1

Answer is (D) exists and is equal to −2.

\((\because\) |(cosx,x,1),(2cosx,2x,2),(tanx,x,1)|=2|(cosx,x,1),(cosx,x,1),(tanx,x,1)|=0)

Comments

If we have to take the determinant anyway after differentiating, why not just take the determinant first and then differentiate? You'd only have to calculate one determinant instead of two.

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Both processes are correct. We also solve the determinant first, then differentiate it to find the solution