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VashaNatasha [74]
3 years ago
8

Tính đạo hàm y= ( x^2+1)(x^3+1)^3

Mathematics
1 answer:
elena-s [515]3 years ago
4 0

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)}

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