What is the d/(dy)((cy(x))/(x^2+y^2)) ?

The d/(dy)((cy(x))/(x^2+y^2)) is (c(x^2-{y}(x)^2))/((x^2+{y)(x)^2)^2}

What is the first d/(dy)((cy(x))/(x^2+y^2)) ?

The first d/(dy)((cy(x))/(x^2+y^2)) is (c(x^2-{y}(x)^2))/((x^2+{y)(x)^2)^2}

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\frac{d}{d y ( x )} (\frac{c y ( x )}{x ^{2} + y ( x ) ^{2}})

\frac{c ( x ^{2} - y ( x ) ^{2} )}{( x ^{2} + y ( x ) ^{2} ) ^{2}}

Solution steps

\frac{d}{d y ( x )} (\frac{c y ( x )}{x ^{2} + y ( x ) ^{2}})

Treat

c, x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= c \frac{d}{d y ( x )} (\frac{y ( x )}{x ^{2} + y ( x ) ^{2}})

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= c \frac{\frac{d}{d y ( x )} ( y ( x ) ) ( x ^{2} + y ( x ) ^{2} ) - \frac{d}{d y ( x )} ( x ^{2} + y ( x ) ^{2} ) y ( x )}{( x ^{2} + y ( x ) ^{2} ) ^{2}}

\frac{d}{d y ( x )} (y (x)) = 1

\frac{d}{d y ( x )} (x^{2} + y (x)^{2}) = 2 y (x)

= c \frac{1 \cdot ( x ^{2} + y ( x ) ^{2} ) - 2 y ( x ) y ( x )}{( x ^{2} + y ( x ) ^{2} ) ^{2}}

Simplify

c \frac{1 \cdot ( x ^{2} + y ( x ) ^{2} ) - 2 y ( x ) y ( x )}{( x ^{2} + y ( x ) ^{2} ) ^{2}} : \frac{c ( x ^{2} - y ( x ) ^{2} )}{( x ^{2} + y ( x ) ^{2} ) ^{2}}

= \frac{c ( x ^{2} - y ( x ) ^{2} )}{( x ^{2} + y ( x ) ^{2} ) ^{2}}

What is the d/(dy)((cy(x))/(x^2+y^2)) ?
The d/(dy)((cy(x))/(x^2+y^2)) is (c(x^2-{y}(x)^2))/((x^2+{y)(x)^2)^2}
What is the first d/(dy)((cy(x))/(x^2+y^2)) ?
The first d/(dy)((cy(x))/(x^2+y^2)) is (c(x^2-{y}(x)^2))/((x^2+{y)(x)^2)^2}