An automatic transmission is generally understood to be any transmission that can change gear ratios by itself without any input from the driver. These days, the lines between manual and automatic transmissions are increasingly blurring, as many architecturally "manual" transmissions use computer controls and electric servos to operate the clutch and change gears. Functionally, we consider these "manumatics" to be automatics. The same is technically true for continuously variable transmissions, or CVTs, since they also change ratios without driver input. In short, if it doesn't require you to use a stick and a clutch to change gears, it's an automatic transmission.
A "transaxle" is a combination of a "transmission" and an "axle." An automatic transaxle starts out as a normal automatic transmission, which bolts to the engine's flywheel via a torque converter and a flexplate. Then there's the transmission itself, which houses all the gears and gear-changing mechanisms. At the "back" of the transmission -- where the driveshaft would enter in a rear-drive application -- lay either a gear or chain drive. This drive connects directly to a differential, which would normally be found inside the the rear axle on a rear-wheel-drive vehicle. The differential unit sends power out the sides toward the front wheels, through the "inner constant velocity joints," to a pair of short axle "half" shafts, then to the "outer CV joints," and finally to the wheels themselves.
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