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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:
Picture <span class=$\alpha$=0">
Tangent vector plot at <span class=$\alpha$=0">

Please help me to understand my mistake in taking dervivative of matrix and multiplying it with image. Thanks in advance. For reference Chapter

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