During which two times of the year (six months apart) does solar altitude and length of day change most rapidly from one day to the next

During the summer and winter solstice.

The summer solstice occurs during June 20th-22nd (in the northern hemisphere) or "the first day of summer". On the day of the summer solstice, the earth's north pole is tilted closest to the sun, and the sun is at the highest point in the sky. The summer solstice is also known as the "longest day of the year" with the most daylight hours.

The winter solstice on the other hand occurs during December 20th-23rd (in the northern hemisphere), or the "first day of winter". On the day of the winter solstice, the earth's north pole is tilted furthest away from sun. The winter solstice is also known as the "shortest day of the year" with the least daylight hours.

Approximately 10 to 14 days after both Equinoxes is when the solar altitude and length of day change most rapidly and the closer to both North & South Poles you get the faster the changes also.I live in Norway at latitude 60 N. Here, we clearly see the difference of the seasons. During the winter, we have perhaps only four hours of daylight. But during the summer, it is the opposite. And, as you know, north of Norway, they have the midnight sun around the summer solstice time.

There can be more freedom for them