Again, I understand how to implicitly differentiate but i got confused trying to do so with this problem and so would like to see how its done. So i'm trying to find what dy/dx is in the end.
d/dx tan(y) * ln (y + x) = d/dx 1
d/dx tan(y) * ln (y + x) = d/dx 1
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d/dx [tan(y) ∙ ln(y + x)] = d/dx[1]
Product rule:
(d/dx[tan(y)] ∙ ln(y + x)) + (tan(y) ∙ d/dx[ln(y + x)]) = 0
(sec²y (dy/dx) ∙ ln(y + x)) + (tan(y) ∙ (1 + dy/dx)/(y + x)) = 0
(dy/dx) sec²y ln(y + x) + (tany + (dy/dx)tany)/(y + x) = 0
(dy/dx) sec²y ln(y + x) + tany/(y + x) + (dy/dx)tany/(y + x) = 0
(dy/dx) sec²y ln(y + x) + (dy/dx)tany/(y + x) = - tany/(y + x)
dy/dx [sec²y ln(y + x) + tany/(y + x)] = - tany/(y + x)
dy/dx = [- tany/(y + x)] / [sec²y ln(y + x) + tany/(y + x)]
Product rule:
(d/dx[tan(y)] ∙ ln(y + x)) + (tan(y) ∙ d/dx[ln(y + x)]) = 0
(sec²y (dy/dx) ∙ ln(y + x)) + (tan(y) ∙ (1 + dy/dx)/(y + x)) = 0
(dy/dx) sec²y ln(y + x) + (tany + (dy/dx)tany)/(y + x) = 0
(dy/dx) sec²y ln(y + x) + tany/(y + x) + (dy/dx)tany/(y + x) = 0
(dy/dx) sec²y ln(y + x) + (dy/dx)tany/(y + x) = - tany/(y + x)
dy/dx [sec²y ln(y + x) + tany/(y + x)] = - tany/(y + x)
dy/dx = [- tany/(y + x)] / [sec²y ln(y + x) + tany/(y + x)]
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tan(y) * ln(y + x) = 1
u = tan(y)
du = sec(y)^2 * dy
v = ln(y + x)
dv = (dy + dx) / (y + x)
udv + vdu = d(1)
u * dv + v * du = 0
tan(y) * (dy + dx) / (y + x) + ln(y + x) * sec(y)^2 * dy = 0
tan(y) * dy / (y + x) + tan(y) * dx / (y + x) + ln(y + x) * sec(y)^2 * dy = 0
dy * (tan(y) / (y + x) + sec(y)^2 * ln(y + x)) = -dx * tan(y) / (y + x)
dy * (1 / (y + x)) * (tan(y) + sec(y)^2 * ln(y + x) * (y + x)) = -dx * tan(y) / (y + x)
dy * (tan(y) + sec(y)^2 * ln(y + x) * (y + x)) = -tan(y) * dx
At this point I'd use some substitution for simplification. For instance ln(y + x) = 1 / tan(y) (look at the original equation to see why)
dy * (tan(y) + sec(y)^2 * (1/tan(y)) * (y + x)) = -tan(y) * dx
dy * (tan(y) + sec(y)^2 * (y + x) / tan(y)) = -tan(y) * dx
dy/dx = -tan(y) / (tan(y) + sec(y)^2 * (y + x) / tan(y))
dy/dx = -tan(y)^2 / (tan(y)^2 + sec(y)^2 * (y + x))
dy/dx = (-sin(y)^2/cos(y)^2) / (sin(y)^2/cos(y)^2 + (1/cos(y)^2) * (y + x))
dy/dx = -sin(y)^2 / (sin(y)^2 + y + x)
u = tan(y)
du = sec(y)^2 * dy
v = ln(y + x)
dv = (dy + dx) / (y + x)
udv + vdu = d(1)
u * dv + v * du = 0
tan(y) * (dy + dx) / (y + x) + ln(y + x) * sec(y)^2 * dy = 0
tan(y) * dy / (y + x) + tan(y) * dx / (y + x) + ln(y + x) * sec(y)^2 * dy = 0
dy * (tan(y) / (y + x) + sec(y)^2 * ln(y + x)) = -dx * tan(y) / (y + x)
dy * (1 / (y + x)) * (tan(y) + sec(y)^2 * ln(y + x) * (y + x)) = -dx * tan(y) / (y + x)
dy * (tan(y) + sec(y)^2 * ln(y + x) * (y + x)) = -tan(y) * dx
At this point I'd use some substitution for simplification. For instance ln(y + x) = 1 / tan(y) (look at the original equation to see why)
dy * (tan(y) + sec(y)^2 * (1/tan(y)) * (y + x)) = -tan(y) * dx
dy * (tan(y) + sec(y)^2 * (y + x) / tan(y)) = -tan(y) * dx
dy/dx = -tan(y) / (tan(y) + sec(y)^2 * (y + x) / tan(y))
dy/dx = -tan(y)^2 / (tan(y)^2 + sec(y)^2 * (y + x))
dy/dx = (-sin(y)^2/cos(y)^2) / (sin(y)^2/cos(y)^2 + (1/cos(y)^2) * (y + x))
dy/dx = -sin(y)^2 / (sin(y)^2 + y + x)
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d/dx( tan(y) * ln (y + x) ) = d/dx( 1)
sec^2y(dy/dx)ln(y + x) + tany(dy/dx + 1)/(y + x) = 0
dy/dx(sec^2y)ln(y + x) + tany(dy/dx)/(y + x) + tany/(y + x) = 0
dy/dx(sec^2y)ln(y + x) + (dy/dx)tany/(y + x) = -tany/(y + x)
dy/dx(sec^2y(ln(y + x)) + tany/(y + x)) = -tany/(y + x)
dy/dx = (-tany/(y + x))/(sec^2y(ln(y + x)) + tany/(y + x))
sec^2y(dy/dx)ln(y + x) + tany(dy/dx + 1)/(y + x) = 0
dy/dx(sec^2y)ln(y + x) + tany(dy/dx)/(y + x) + tany/(y + x) = 0
dy/dx(sec^2y)ln(y + x) + (dy/dx)tany/(y + x) = -tany/(y + x)
dy/dx(sec^2y(ln(y + x)) + tany/(y + x)) = -tany/(y + x)
dy/dx = (-tany/(y + x))/(sec^2y(ln(y + x)) + tany/(y + x))