implicit derivative of (d^2y)/(dx^2), y=sin(3x+4y)

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implicit derivative d²ydx² , y=sin(3x+4y)

Solution

d²ydx² =−9sin(3x+4y)(1−4cos(3x+4y))³

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y=sin(3x+4y)

Treat y as y(x)

Implicit Derivative dydx of y=sin(3x+4y): 3cos(3x+4y)1−4cos(3x+4y)

dydx =3cos(3x+4y)1−4cos(3x+4y)

Implicit Derivative d²ydx² of dydx =3cos(3x+4y)1−4cos(3x+4y) : 3(−3sin(3x+4y)−4sin(3x+4y)dydx )(1−4cos(3x+4y))²

d²ydx² =3(−3sin(3x+4y)−4sin(3x+4y)dydx )(1−4cos(3x+4y))²

Substitute dydx =3cos(3x+4y)1−4cos(3x+4y) : −9sin(3x+4y)(1−4cos(3x+4y))³

d²ydx² =−9sin(3x+4y)(1−4cos(3x+4y))³

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+▭▭	>	4	5	6	×	☐☐☐
×▭▭	(	1	2	3	−	`x`
▭ \|▭	)	.	0	=	+	`y`

implicit derivative of (d^2y)/(dx^2), y=sin(3x+4y)

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