1) 2x⁶y³ - ln7y*ln8x=0;
(2x⁶y³ - ln7y*ln8x)'=0;
12x⁵y³ + 2x⁶·3y²·y' - (1/7y)·7y'·ln8x - ln7y·(1/8x)·8 = 0;
12x⁵y³ + 6x⁶y²·y' - (y'ln8x)/y - ln7y·(1/x) = 0;
12x⁵y³ + y'(6x⁶y² - (ln8x)/y) - ln7y·(1/x) = 0;
y'(6x⁶y² - (ln8x)/y) = ln7y/x - 12x⁵y³;
y' = (ln7y/x - 12x⁵y³)/(6x⁶y² - (ln8x)/y).
2) y=(5x²+6)^arctg8x;
lny = arctg8x·ln(5x²+6);
(lny)' = (arctg8x·ln(5x²+6))';
y'/y = (8/(1+64x²))·ln(5x²+6) + (10x·arctg8x)/(5x²+6);
y' = y((8/(1+64x²))·ln(5x²+6) + (10x·arctg8x)/(5x²+6));
y' = ((5x^2+6)^arctg8x)·((8/(1+64x²))·ln(5x²+6) + (10x·arctg8x)/(5x²+6));
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Answers & Comments
1) 2x⁶y³ - ln7y*ln8x=0;
(2x⁶y³ - ln7y*ln8x)'=0;
12x⁵y³ + 2x⁶·3y²·y' - (1/7y)·7y'·ln8x - ln7y·(1/8x)·8 = 0;
12x⁵y³ + 6x⁶y²·y' - (y'ln8x)/y - ln7y·(1/x) = 0;
12x⁵y³ + y'(6x⁶y² - (ln8x)/y) - ln7y·(1/x) = 0;
y'(6x⁶y² - (ln8x)/y) = ln7y/x - 12x⁵y³;
y' = (ln7y/x - 12x⁵y³)/(6x⁶y² - (ln8x)/y).
2) y=(5x²+6)^arctg8x;
lny = arctg8x·ln(5x²+6);
(lny)' = (arctg8x·ln(5x²+6))';
y'/y = (8/(1+64x²))·ln(5x²+6) + (10x·arctg8x)/(5x²+6);
y' = y((8/(1+64x²))·ln(5x²+6) + (10x·arctg8x)/(5x²+6));
y' = ((5x^2+6)^arctg8x)·((8/(1+64x²))·ln(5x²+6) + (10x·arctg8x)/(5x²+6));