**** keep in mind (dy/dx) becomes like a letter itself
5.
x⁴+y⁴=2
Taking the derivative of both sides with respect to x:
4x³ + 4y³ (dy/dx) = 0
4y³ (dy/dx) = -4x³
(dy/dx) = -4x³/4y³
4's cancel out
(dy/dx) = -x³/y³
At (1,-1), we have
(dy/dx) = -x³/y³
(dy/dx) = -(1)³/(-1)³
(dy/dx) = -1/-1
(dy/dx) = 1
7.
y² = 4x
Taking the derivative of both sides with respect to x:
2y (dy/dx) = 4
divide 2y on both sides
(dy/dx) = 4/2y
simplify
(dy/dx) = 2/y
At (1,2), we have
(dy/dx) = 2/y
(dy/dx) = 2/(2)
(dy/dx) = 1
9.
sin y=5x⁴-5
Taking the derivative of both sides with respect to x:
cos y (dy/dx) = 20x³
(dy/dx) = 20x³/cos y
At (1,π), we have (dy/dx) = 20/(-1) = -20.
11.
cos y=x
Taking the derivative of both sides with respect to x:
-sin y (dy/dx) = 1
(dy/dx) = -1/sin y
At (0,π/2), we have (dy/dx) = -1.
6.
x=e^y
Taking the derivative of both sides with respect to x:
1 = e^y (dy/dx)
(dy/dx) = 1/e^y
At (2,In 2), we have (dy/dx) = 1/e^(In 2) = 1/2.
8.
y²+3x= 8
Taking the derivative of both sides with respect to x:
2y (dy/dx) + 3 = 0
(dy/dx) = -3/(2y)
At (1,√5), we have (dy/dx) = -3/(2√5).
10.
√x-2√y = 0
Taking the derivative of both sides with respect to x:
1/(2√x) - 1/√y (dy/dx) = 0
(dy/dx) = √y/(2√x)
At (4,1), we have (dy/dx) = 1/4.
12.
tan xy=x+y
Taking the derivative of both sides with respect to x:
y sec² (xy) (dy/dx) = 1 + y
(dy/dx) = (1 + y)/[y sec² (xy)]
At (0,0), we have (dy/dx) = 1/0, which is undefined.
5-12. Implicit differentiation Carry out the following steps.
a. Use implicit differentiation to find (dy/dx)
b. Find the slope of the curve at the given point.
5. x⁴+y⁴=2 ; (1,-1)
6. y² = 4x ; (1,2)
7. sin y=5x⁴-5 ; (1,π)
8. cos y=x ; (0, (π/2))
9. x=e^y ; (2, In 2)
10. y²+3x= 8 ; (1, √5)
11. √x-2√y = 0 ; (4,1)
12. tan xy=x+y ; (0,0)
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