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I have a question regarding first order derivatives. If we consider the function $f : \mathbb{R}^2\rightarrow\mathbb{R}^3$ with $(x_1, x_2)\mapsto (x^2_1+ x_2, x^3_2, \cos{x_2})$. We have to calculate $f'(0)(a)$ and $f''(0)(a)(b)$ where $a = (a_1, a_2)$ and $b = (b_1, b_2).$

Can some one please tell me how one is supposed to tackle this problem. Thanks.

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    Do you mean $f ′(0)(a)(b)$ or simply $f ′(0)(b)$? Only the latter makes sense to me! So I may assume the latter. Just differentiate each co-ordinate w.r.t. x_1, x_2 and write down the derivatives w.r.t. x_i of each function in one row i.e., $$\begin{pmatrix}\frac{df_1}{dx_1}&\frac{df_1}{dx_2}\\\frac{df_2}{dx_1}&\frac{df_2}{dx_2}\\\frac{df_3}{dx_1}&\frac{df_3}{dx_2}\end{pmatrix}$$ Now evaluate at $0$ and left-multiply $a$ and $b$ separately with the resulting $3\times2$ matrix.2012-07-10
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    Evaluating at 0 gives the 3 x 2 matrix (0 1;0 0; 0 0). Hence, the derivatives at a is simply (a2 0 0). Is that correct? Also, it is f '' (0) (a) (b).2012-07-10

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