Hallar por definición la siguiente derivada?

<img src="https://tex.z-dn.net/?f=x%28t%29%3D%5Cdfrac%7BAt%2BB%7D%7BCt%2BD%7D" />

Recordemos la definición de la derivada (solo la parte que nos interesa)

<img src="https://tex.z-dn.net/?f=%5Cdfrac%7Bdx%7D%7Bdt%7D%3D%5Clim%5Climits_%7Bt%5Cto%20t_0%7D%5Cdfrac%7Bx%28t%29-x%28t_0%29%7D%7Bt-t_0%7D%5C%5C%20%5C%5C%0A%5Ctexttt%7Bo%7D%5C%5C%20%5C%5C%20%0A%5Cdfrac%7Bdx%7D%7Bdt%7D%3D%5Clim%5Climits_%7Bh%5Cto%200%7D%5Cdfrac%7Bx%28h%2Bt_0%29-x%28t_0%29%7D%7Bh%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BEntonces%7D%5C%5C%20%5C%5C%0A%5Cdfrac%7Bdx%7D%7Bdt%7D%3D%5Clim%5Climits_%7Bh%5Cto%200%7D%5Cdfrac%7B%5Cdfrac%7BA%28h%2Bt_0%29%2BB%7D%7BC%28h%2Bt_0%29%2BD%7D-%5Cdfrac%7BAt_0%2BB%7D%7BCt_0%2BD%7D%7D%7Bh%7D" />

<img src="https://tex.z-dn.net/?f=%5Cdfrac%7Bdx%7D%7Bdt%7D%3D%5Clim%5Climits_%7Bh%5Cto%200%7D%5Cdfrac%7B%20%5BA%28h%2Bt_0%29%2BB%5D%28Ct_0%2BD%29-%28At_0%2BB%29%5BC%28h%2Bt_0%29%2BD%5D%7D%7Bh%5BC%28h%2Bt_0%29%2BD%5D%28Ct_0%2BD%29%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ctexttt%7BHaciendo%20operaciones%20en%20el%20numerador...%7D%5C%5C%20%5C%5C%0A%5Cdfrac%7Bdx%7D%7Bdt%7D%3D%5Clim%5Climits_%7Bh%5Cto%200%7D%5Cdfrac%7B%20h%28AD-BC%29%7D%7Bh%5BC%28h%2Bt_0%29%2BD%5D%28Ct_0%2BD%29%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cdfrac%7Bdx%7D%7Bdt%7D%3D%5Clim%5Climits_%7Bh%5Cto%200%7D%5Cdfrac%7B%20AD-BC%7D%7B%5BC%28h%2Bt_0%29%2BD%5D%28Ct_0%2BD%29%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cdfrac%7Bdx%7D%7Bdt%7D%3D%5Cdfrac%7B%20AD-BC%7D%7B%28Ct_0%2BD%29%28Ct_0%2BD%29%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Cdfrac%7Bdx%7D%7Bdt%7D%3D%5Cdfrac%7B%20AD-BC%7D%7B%28Ct_0%2BD%29%5E2%7D%5C%5C%20%5C%5C%20%5C%5C" />

Observación : entiéndase a<img src="https://tex.z-dn.net/?f=%5Cdfrac%7Bdx%7D%7Bdt%7D" /> como la derivada de la función<img src="https://tex.z-dn.net/?f=x%3AD%5Csubset%20%5Cmathbb%20R%20%5Clongrightarrow%20%5Cmathbb%20R" /> en el punto<img src="https://tex.z-dn.net/?f=x%3Dx_0%5Cin%20D" />, donde<img src="https://tex.z-dn.net/?f=D%20" /> es un conjunto abierto en el cual la función susodicha es derivable.