§ 37: 37.14 (1-4)

37.14 Знайдіть похідну функції:

1) y’ = x’ • 2x+1 + x • (2x+1)’ = 2x+1 + x22x+1 • (2x + 1)’ = 2x+1 + x • 222x+1 = (2x+1)2+ x2x+1 = 2x+1+x2x+1 = 3x+12x+1;

2) y’ = (sinx)’cos2x + sinx • (cos2x)’ = cosx • cos2x + sinx • (–sin2x) • 2 = cosxcos2x – 2sinxsin2x;

3) y’ = (tgx)’ • sin(2x + 5) + tgx • (sin(2x + 5)) = 1cos2x • sin(2x + 5) + tg x • cos(2x + 5) • 2 = sin⁡(2x+5)cos2x + 2 tg x • cos(2x + 5);

4) y’ = (cos3x)' • (x-1)-cos⁡3x • (x-1)' (x-1)2 = -sin3x • 3x-1-cos3x(x-1)2 = -3x-1sin3x-cos3x(x-1)2;

5) y’ = x- 1'x+ 1- (x- 1)(x+ 1)'(x+ 1)2 = x+ 12x- x- 12x(x+ 1)2 = x+ 1- x+ 12x (x+ 1)2 = 1x(x+ 1)2;

6) y’ = x2+ 1 • x-x2+ 1 • x'x2 = x • 2x2x2+ 1- x2+ 1x2 = 2x2- 2(x2+ 1)22x2x2+ 1 = 2x2- 2x2- 22x2x2+ 1 = –1x2x2+ 1.