(A) Sin(y/x) +ye^(2-x) = x-y (B) √y - 4ln x = cos (x+y) (C) e^(2y) + tan(1/x) =(x²/y) (D) ln y+2x =(3-y)^3 (E)xy²-sin(x+2y)=2x . Find the dy/dx (implicit differentiation)?
1 Answer
To keep the answer at a reasonable length, I will only show you parts (A) and (E).
Explanation:
So, for part (A) you have
sin(y/x) + ye^(2-x) = x- ysin(yx)+ye2−x=x−y
To get
d/dx[sin(y/x) + ye^(2-x)] = d/dx(x-y)ddx[sin(yx)+ye2−x]=ddx(x−y)
To make the calculations easier to follow, I'll solve each of these derivatives separately. First, you have
d/dx[sin(y/x)] = cos(y/x) * d/dx(y * x^(-1))ddx[sin(yx)]=cos(yx)⋅ddx(y⋅x−1)
d/dx[sin(y/x)] = cos(y/x) * [(dy)/dx * x^(-1) + y * (-x)^(-2)]ddx[sin(yx)]=cos(yx)⋅[dydx⋅x−1+y⋅(−x)−2]
Next
d/dx(ye^(2-x)) = (dy)/dx * e^(2-x) + y * d/dx(e^(2-x))ddx(ye2−x)=dydx⋅e2−x+y⋅ddx(e2−x)
d/dx(ye^(2-x)) = (dy)/dx * e^(2-x) + y * e^(2-x) * (-1)ddx(ye2−x)=dydx⋅e2−x+y⋅e2−x⋅(−1)
d/dx(ye^(2-x)) = (dy)/dx * e^(2-x) - y * e^(2-x)ddx(ye2−x)=dydx⋅e2−x−y⋅e2−x
Plug these back into your target calculation to get
cos(y/x) * (dy)/dx * x^(-1) + cos(y/x) * (-yx^(-2)) + (dy)/dx * e^(2-x) - y * e^(2-x) = 1 - (dy)/dxcos(yx)⋅dydx⋅x−1+cos(yx)⋅(−yx−2)+dydx⋅e2−x−y⋅e2−x=1−dydx
Isolate
cos(y/x) * (dy)/dx * x^(-1) + (dy)/dx * e^(2-x) + (dy) = 1 + y * e^(2-x) +yx^(-2) * cos(y/x)cos(yx)⋅dydx⋅x−1+dydx⋅e2−x+(dy)=1+y⋅e2−x+yx−2⋅cos(yx)
(dy)/dx * [x^(-1)cos(y/x) + e^(2-x) + 1] = 1 + y * e^(2-x) +yx^(-2) * cos(y/x)dydx⋅[x−1cos(yx)+e2−x+1]=1+y⋅e2−x+yx−2⋅cos(yx)
Finally, you have
(dy)/dx = [1 + y * e^(2-x) +yx^(-2) * cos(y/x)]/[1 + e^(2-x) - x^(-1)cos(y/x)]dydx=1+y⋅e2−x+yx−2⋅cos(yx)1+e2−x−x−1cos(yx)
You can simplify this to the form
(dy)/dx = (x^2 + y * x^2 * e^(2-x) + ycos(y/x))/x^color(red)(cancel(color(black)(2))) * color(red)(cancel(color(black)(x)))/(x + x * e^(2-x) - cos(y/x))
(dy)/dx = color(green)((x^2 + y x^2e^(2-x) + ycos(y/x))/(x[x + xe^(2-x) - cos(y/x)])
For part (E) you have
xy^2 - sin(x+ 2y) = 2x
The exact same approach applies here as well. Differentiate both sides with respect to
d/dx[xy^2 - sin(x + 2y)] = d/dx(2x)
This will get you
d/dx(xy^2) = d/dx(x) * y^2 + x * (dy)/dx * 2y
d/dx(xy^2) = y^2 + 2xy(dy)/dx
and
d/dx[sin(x + 2y)] = cos(x + 2y) * d/dx(x + 2y)
d/dx[sin(x + 2y)] = cos(x + 2y) * [d/dx(x) + 2(dy)/dx]
d/dx[sin(x + 2y)] = cos(x + 2y) * (1 + 2(dy)/dx)
Plug these back into your target calculation to get
y^2 + 2xy(dy)/dx - [cos(x + 2y) + 2 cos(x + 2y)(dy)/dx] = 2
y^2 + 2xy(dy)/dx - cos(x + 2y) - 2cos(x + 2y)(dy)/dx = 2
Once again, isolate
(dy)/dx * 2[xy - cos(x + 2y)] = 2 - y^2 + cos(x + 2y)
(dy)/dx = color(green)((2 - y^2 + cos(x + 2y))/(2[xy - cos(x + 2y)])
