35. Which of the following functions is an explicit function?
(1) sin(x + y) = xy
(2) ex+y = x2 - 3x + 2
(3) log(x + y) = COSX + Cosy
(4) 3* = 5x + y(x2 + xy + y)
Answers
Answer:
Step-by-step explanation:
Let's differentiate x^2+y^2=1x
2
+y
2
=1x, squared, plus, y, squared, equals, 1 for example. Here, we treat yyy as an implicit function of xxx.
\begin{aligned} x^2+y^2&=1 \\\\ \dfrac{d}{dx}(x^2+y^2)&=\dfrac{d}{dx}(1) \\\\ \dfrac{d}{dx}(x^2)+\dfrac{d}{dx}(y^2)&=0 \\\\ 2x+2y\cdot\dfrac{dy}{dx}&=0 \\\\ 2y\cdot\dfrac{dy}{dx}&=-2x \\\\ \dfrac{dy}{dx}&=-\dfrac{x}{y} \end{aligned}
x
2
+y
2
dx
d
(x
2
+y
2
)
dx
d
(x
2
)+
dx
d
(y
2
)
2x+2y⋅
dx
dy
2y⋅
dx
dy
dx
dy
=1
=
dx
d
(1)
=0
=0
=−2x
=−
y
x
Notice that the derivative of y^2y
2
y, squared is 2y\cdot\dfrac{dy}{dx}2y⋅
dx
dy
2, y, dot, start fraction, d, y, divided by, d, x, end fraction and not simply 2y2y2, y. This is because we treat yyy as a function of xxx.
Step-by-step explanation:
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