Question

Tính đạo hàm y= ( x^2+1)(x^3+1)^3

Answer 1

Use the product, power, and chain rules.

$y = (x^2 + 1) (x^3 + 1)^3$

Differentiate both sides:

$\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{\mathrm d\left((x^2 + 1) (x^3 + 1)^3\right)}{\mathrm dx}$

Product rule:

$\dfrac{\mathrm dy}{\mathrm dx} = \dfrac{\mathrm d(x^2 + 1)}{\mathrm dx}(x^3+1)^3 + (x^2+1)\dfrac{\mathrm d(x^3 + 1)^3}{\mathrm dx}$

Power rule for the first derivative, power and chain rules for the second one:

$\dfrac{\mathrm dy}{\mathrm dx} = 2x(x^3+1)^3 + 3(x^2+1)(x^3+1)^2\dfrac{\mathrm d(x^3 + 1)}{\mathrm dx}$

One last applicaton of power rule:

$\dfrac{\mathrm dy}{\mathrm dx} = 2x(x^3+1)^3 + 9x^2(x^2+1)(x^3+1)^2$

You could stop here, or continue and simplify the result by factorizing:

$\dfrac{\mathrm dy}{\mathrm dx} = x(x^3+1)^2 \left(2(x^3+1) + 9x(x^2+1)\right) \\\\ \dfrac{\mathrm dy}{\mathrm dx} = \boxed{x(x^3+1)^2 (11x^3+9x+2)}$