# Calculating tangent vector of curve s(P,$\alpha$) at given point $\alpha$ = 0. http://yann.lecun.com/exdb/publis/pdf/simard-00.pdf

I am reading one chapter where tangent vector is calculated for the given curve $$s(P,\alpha)$$ at $$\alpha=0$$ by differentiating with respect to $$\alpha$$; $$\frac{\partial s(P,\alpha)}{\partial\alpha}$$. For the curve I have taken one $$2D$$ image and I am rotating it with matrix $$R=[\matrix{cos(\alpha)\space -sin(\alpha)\\sin(\alpha) \space\space\space\space\space cos(\alpha)}]$$.

As my image is fixed, therefore the curve is just a function of $$\alpha$$. Therefore to find the tangent vector what I am doing is as follows:
1. I am rotating the image by the matrix $$R^{'}$$ which is $$R^{'}=[\matrix{-sin(\alpha)\space -cos(\alpha)\\cos(\alpha) \space\space\space\space\space -sin(\alpha)}]$$
2.This rotates the image by $$90$$ degree, which is not the expected result.

I have done the same exercise by differentiating numerically and I am getting the expected answer which is as follows:
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Please help me to understand my mistake in taking dervivative of matrix and multiplying it with image. Thanks in advance. For reference Chapter