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Popular >

(d^3)/(dx^3)((y)(x^3+3x^2+6x))

Solution

\frac{d ^{3}}{d x ^{3}} ((y) (x^{3} + 3 x^{2} + 6 x))

Solution

x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 6 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 18 x \frac{d y}{d x} + 18 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 6 y

Solution steps

\frac{d ^{3}}{d x ^{3}} ((y) (x^{3} + 3 x^{2} + 6 x))

\frac{d}{d x} (y (x^{3} + 3 x^{2} + 6 x)) = \frac{d y}{d x} (x^{3} + 3 x^{2} + 6 x) + (3 x^{2} + 6 x + 6) y

= \frac{d ^{2}}{d x ^{2}} (\frac{d y}{d x} (x^{3} + 3 x^{2} + 6 x) + (3 x^{2} + 6 x + 6) y)

\frac{d}{d x} (\frac{d y}{d x} (x^{3} + 3 x^{2} + 6 x) + (3 x^{2} + 6 x + 6) y) = \frac{d ^{2} y}{d x ^{2}} (x^{3} + 3 x^{2} + 6 x) + 2 \frac{d y}{d x} (3 x^{2} + 6 x + 6) + (6 x + 6) y

= \frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{3} + 3 x^{2} + 6 x) + 2 \frac{d y}{d x} (3 x^{2} + 6 x + 6) + (6 x + 6) y)

\frac{d}{d x} (\frac{d ^{2} y}{d x ^{2}} (x^{3} + 3 x^{2} + 6 x) + 2 \frac{d y}{d x} (3 x^{2} + 6 x + 6) + (6 x + 6) y) = x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 6 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 18 x \frac{d y}{d x} + 18 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 6 y

= x^{3} \frac{d ^{3} y}{d x ^{3}} + 3 x^{2} \frac{d ^{3} y}{d x ^{3}} + 9 x^{2} \frac{d ^{2} y}{d x ^{2}} + 6 x \frac{d ^{3} y}{d x ^{3}} + 18 x \frac{d ^{2} y}{d x ^{2}} + 18 x \frac{d y}{d x} + 18 \frac{d ^{2} y}{d x ^{2}} + 18 \frac{d y}{d x} + 6 y

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Frequently Asked Questions (FAQ)

What is (d^3)/(dx^3)((y)(x^3+3x^2+6x)) ?
The answer to (d^3)/(dx^3)((y)(x^3+3x^2+6x)) is