What is (d^2)/(dx^2)((y)(x^m)) ?

The answer to (d^2)/(dx^2)((y)(x^m)) is x^m(d^2y)/(dx^2)+m^2x^{m-2}y+2mx^{m-1}(dy)/(dx)-mx^{m-2}y

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(d^2)/(dx^2)((y)(x^m))

\frac{d ^{2}}{d x ^{2}} ((y) (x^{m}))

x^{m} \frac{d ^{2} y}{d x ^{2}} + m^{2} x^{m - 2} y + 2 m x^{m - 1} \frac{d y}{d x} - m x^{m - 2} y

Solution steps

\frac{d ^{2}}{d x ^{2}} ((y) (x^{m}))

\frac{d}{d x} (y x^{m}) = \frac{d y}{d x} x^{m} + m x^{m - 1} y

= \frac{d}{d x} (\frac{d y}{d x} x^{m} + m x^{m - 1} y)

\frac{d}{d x} (\frac{d y}{d x} x^{m} + m x^{m - 1} y) = x^{m} \frac{d ^{2} y}{d x ^{2}} + m^{2} x^{m - 2} y + 2 m x^{m - 1} \frac{d y}{d x} - m x^{m - 2} y

= x^{m} \frac{d ^{2} y}{d x ^{2}} + m^{2} x^{m - 2} y + 2 m x^{m - 1} \frac{d y}{d x} - m x^{m - 2} y

x^{3} + 2 x + 5 x + 10

a^{2} x + a^{2} y + b^{2} x + b^{2} y

[11]

x^{2} + 2^{3}

(x^{2}) + 4

What is (d^2)/(dx^2)((y)(x^m)) ?
The answer to (d^2)/(dx^2)((y)(x^m)) is x^m(d^2y)/(dx^2)+m^2x^{m-2}y+2mx^{m-1}(dy)/(dx)-mx^{m-2}y