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I need help understanding this proof: http://mathworld.wolfram.com/Curvature.html" [Broken]

I understand everything from step 1 to step 7. However I don't understand how the result of step 7 is applied in step 8 and in step 9.

I understand that

[tex]\tan \phi=\frac{y'}{x'}[/tex] (step 7)

and that

[tex]\frac{\mathrm{d} }{\mathrm{d} t}\tan \phi=\sec^{2}\phi\frac{\mathrm{d} \phi}{\mathrm{d} t}[/tex] (step 8)

But I don't understand how [tex]\tan \phi=\frac{y'}{x'}[/tex] is applied to [tex]\frac{\mathrm{d} }{\mathrm{d} t}\tan \phi=\sec^{2}\phi\frac{\mathrm{d} \phi}{\mathrm{d} t}[/tex] to get [tex]\frac{x'y''-y'x''}{x'^{2}}[/tex]

Same for step 9

Can anyone help me?

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# Need help understanding a proof

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