Question

Using the first rule of derivative find 8(x-y)12?? And how can i solve these y^x+x^y=17,x+y=5

Answer 1

The first part is somewhat vague.

Are you finding the derivative of 8(x-y)^12 with respect to x , with respect to y, with respect to ?

assuming "with respect to x",

d(8(x-y)^12)/dx = 96(x-y)^11 (1 - dy/dx)

your second question:
y^x + x^y = 17 and x+y=5

Since this would be a very nasty question, my first impulse was to assume there might be an integer solution.
So we would have the sum of two powers equal to 17
possible powers ≤ 17
1 --- > anybase^0
4 ---> 2^2
8 ---> 2^3
9 ---> 3^2
16 ---> 4^2

sure enough!!
2^3 + 3^2 = 8+9 = 17 AND 3+2 = 5

so x=2, y=3 OR x=3, y=2