If f(x)=vmatrix x & x & 12 x & x^2 & 2x x & x & 1vmatrix, then _x0f'(x)x

Question

If $f(x)=\begin{vmatrix}\cos x & x & 1\2\sin x & x^2 & 2x\	an x & x & 1\end{vmatrix}$, then $\displaystyle\lim_{x	o0}\dfrac{f'(x)}{x}$

Accepted Answer

Expanding the determinant, $f(x)=x^2(\tan x-\cos x)$.

$$f'(x)=2x(\tan x-\cos x)+x^2(\sec^2x+\sin x)$$
$$\lim_{x\to0}\frac{f'(x)}{x}=\lim_{x\to0}2(\tan x-\cos x)+x(\sec^2x+\sin x)=2(0-1)+0=-2$$

Solutions · 1