What is (\partial)/(\partial x)((4x-5y)/(4x+5y)) ?

The answer to (\partial)/(\partial x)((4x-5y)/(4x+5y)) is (40y)/((4x+5y)^2)

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(\partial)/(\partial x)((4x-5y)/(4x+5y))

\frac{\partial}{\partial x} (\frac{4 x - 5 y}{4 x + 5 y})

\frac{40 y}{( 4 x + 5 y ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{4 x - 5 y}{4 x + 5 y})

Treat

y

as a constant

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial x} ( 4 x - 5 y ) ( 4 x + 5 y ) - \frac{\partial}{\partial x} ( 4 x + 5 y ) ( 4 x - 5 y )}{( 4 x + 5 y ) ^{2}}

\frac{\partial}{\partial x} (4 x - 5 y) = 4

\frac{\partial}{\partial x} (4 x + 5 y) = 4

= \frac{4 ( 4 x + 5 y ) - 4 ( 4 x - 5 y )}{( 4 x + 5 y ) ^{2}}

Expand

4 (4 x + 5 y) - 4 (4 x - 5 y) : 40 y

= \frac{40 y}{( 4 x + 5 y ) ^{2}}

\frac{4}{( s ) ( s ^{2} + 2 s + 4 )}

y^{^{''}} - 2 y^{^{'}} + y = \frac{( - 2 e ^{- t} )}{( t ^{2} + 1 )}

(x^{2} + y^{2} + x) d x + y d y = 0

\frac{\partial}{\partial y} (e^{x + y} - 1)

f (x) = 9 x + 8

What is (\partial)/(\partial x)((4x-5y)/(4x+5y)) ?
The answer to (\partial)/(\partial x)((4x-5y)/(4x+5y)) is (40y)/((4x+5y)^2)