Why choose trained than taught? (PTE)

1. Speed vs Rate:

The questions come down to, which one is correct: "technological change speed" or "technological change rate". Based on Oxford dictionary, rate means:

A measure, quantity, or frequency, typically one measured against another quantity or measure.

The text compares current technological improvement speed to the historical speed. Because there is a comparison (of speed), "rate" is more relevant than "speed".

You can also check some examples in this question, like:

You use "economic growth rate" not "economic growth speed" when it is expressed in a percentage.

2. Trained vs Taught:

Based on this answer:

Usage of the second object differs:

You teach a soccer player an individual skill.

You train a soccer player in a skill. (You can't train someone a skill)

it should be "trained" in this text because there is an "in" after the blanked-out word.

I "Rate" expresses precisely the mathematical idea of ratio in questions where the distance covered by an object is of interest, a ratio being often used in current English to show the importance or the distribution of a quantity in reference to another.

• 10 miles per hour (10 mph)
• 100 pounds per capita (£100 per capita)
• 150 days per year
• 52 milligrams per liter (52 mg/l)

When you refer to speed, the most basic meaning is that of rate of distance covered per unit time; however, "speed" has taken on a figurative sense in which the rate is the number of almost anything per unit time, and the unit of time is very often not even specified in this usage; the unit is understood to be one that makes enough sense. There is no logical reason in your example for choosing "rate" rather than "speed" except a concern for reverting to a purer expression in which analogy has a much less important place.

When the ratio is a rate, or equivalently, a speed (ref.), one is talking about units of distance per unit time, but both words have come to mean "units of < anything > per unit time". As the reference shows "rate" and "speed" are synonyms, but not perfect synonyms insofar as usage is concerned. You say usually "he went to the other side of town at the speed of fifty miles per hour" but if "he went to the other side of town at the rate of fifty miles per hour" is correct technically it is not at all the usual way to say that (you never hear it). Similarly, for a computer, you will speak of the processing speed of the machine rather than the processing rate (ngram) although you can still use this latter term.
Nevertheless, in your example the teachers are trying to give usage a particular turn; this is so because "speed" is still defined as follows and is a synonym of "rate" (you could as well have said "pace", so, you are not wrong).

(OALD, 2) the rate at which something happens or is done — the processing speed of the computer

II "Train" is much more usual than "teach" in the fields of practical skills. "Teach" is usual in talking about lessons in schools and colleges. "Taught" is correct technically (but not grammatically), as it is used to define "train" in dictionaries, but not as idiomatic.
However, the primary reason for not using "teach" is grammar: "in" is a usual preposition for introducing a complement of "to train", whereas it is not one for "to teach".

I agree with the answer by @Kemal Cengiz and have upvoted it.

My answer comes from a slightly different angle but does not contradict anything in the previous answer.

Trained vs Taught

As a rule of thumb, We train a person but we teach a skill. (direct object)

Now add the indirect object

We train a person in a skill.

We teach a skill to a person.

Please note that this is not a cast-iron rule but it may give you an insight into this particular case - see note at end for finer details.

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Speed vs Rate

Speed is a rate, however a rate is not necessarily a speed. Usually speed is the rate of change of position with time. Rate can be the change of anything with respect to time. Some would argue that rate can be the change of anything with respect to anything. Thus we might talk about the rate of change of volume with respect to pressure, (time doesn't even come into the equation). Also speed is much less well defined than rate, for example, we can say

He drove at speed to get to the hospital.

but we cannot say

He drove at rate to get to the hospital.

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Note about train vs teach

In fact we can teach a skill and we can teach a person. We do not teach "in" something.

However while we can train a person, we rarely, if ever, talk about training a skill. We train a person in a skill or to do/perform a skill.

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P.S. This answer was written before the one by @LPH but wasn't submitted until afterwards (the plumber arrived!)