§ 37: 37.3-37.4

37.3 Знайдіть похідну функції:

1) y’ = (x + 2)' • (x² – 4x + 5) + (x + 2)(х² – 4х + 5)’ = х² – 4х + 5 + (x + 2)(2х – 4) = x² – 4х + 5 + 2х² – 4х + 2х – 8 = Зх² – 6x – 3;

2) y’ = (Зх + 5)(2х² – 1) + (Зх + 5)(2х² – 1)’ – 3 • (2х² – 1) + (Зх + 5) • 4х = 6х² – 3 + 12х² + 20х = 18х² + 20х – 3;

3) y’ – (x2)’ • sinx + x² • (sinх)’ = 2х • sinx + x²cosx = 2хsinx + x²cosх;

4) у’ = х’ • ctgx + х • (ctgx)’ = 1 • ctgx + x • (–1sin2x) = ctgx – xsin2x;

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

6) y’ = (x)’ cos x + x • (cos x)’ = 12x • cos x + x • (–sinx) = cosx2x – xsinx.

37.4 Знайдіть похідну функції:

1) y’ = (x³ – 2)’ (x² + 1) + (x³ – 2) • (x² + 1)’ = 3x²(x² + 1) + (x³ – 2) • 2x = 3x⁴ + 3x² + 2x4 – 4x = 5x⁴ + 3x² – 4x;

2) y’ = (x + 5)’ • x + (x + 5) • (x)’ = x + (x + 5) • 12x = 2x • x+ x+52x = 2x+x+52x = 3x+52x;

3) y’ = (x⁴)’cosx + x⁴(cosx)’ = 4x³cosx + x⁴(–sinx) = 4x³cosx – x⁴sinx;

4) y’ = x’tgx + x(tgx)’ = tgx + xcos2x.