What is the derivative of ((φ(x^2+2y(φ,x)+ψ(φ,x,y(φ,x))(x-2y(φ,x)^2)))/x) ?

The derivative of ((φ(x^2+2y(φ,x)+ψ(φ,x,y(φ,x))(x-2y(φ,x)^2)))/x) is

What is the first derivative of ((φ(x^2+2y(φ,x)+ψ(φ,x,y(φ,x))(x-2y(φ,x)^2)))/x) ?

The first derivative of ((φ(x^2+2y(φ,x)+ψ(φ,x,y(φ,x))(x-2y(φ,x)^2)))/x) is

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\frac{d}{d x} \frac{( φ ( x ^{2} + 2 y ( φ , x ) ) + ψ ( φ , x , y ( φ , x ) ) ( x - 2 y ( φ , x ) ( φ , x ) ^{2} ) )}{x}

\frac{( φ ( 2 x + 2 \frac{d}{d x} ( y ( φ , x ) ) ) + ( \frac{d}{d x} ( ψ ( φ , x , y ( φ , x ) ) ) ( x - 2 y ( φ , x ) ( φ , x ) ^{2} ) + ( 1 - 2 ( \frac{d}{d x} ( y ( φ , x ) ) ( φ , x ) ^{2} + 2 ( φ , x ) ( φ , 1 ) y ( φ , x ) ) ) ψ ( φ , x , y ( φ , x ) ) ) ) x - 1 \cdot ( φ ( x ^{2} + 2 y ( φ , x ) ) + ψ ( φ , x , y ( φ , x ) ) ( x - 2 y ( φ , x ) ( φ , x ) ^{2} ) )}{x ^{2}}

Solution steps

\frac{d}{d x} \frac{( φ ( x ^{2} + 2 y ( φ , x ) ) + ψ ( φ , x , y ( φ , x ) ) ( x - 2 y ( φ , x ) ( φ , x ) ^{2} ) )}{x}

Treat

φ

as a constant

Apply the Quotient Rule:

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

= \frac{\frac{d}{d x} ( φ ( x ^{2} + 2 y ( φ , x ) ) + ψ ( φ , x , y ( φ , x ) ) ( x - 2 y ( φ , x ) ( φ , x ) ^{2} ) ) x - \frac{d x}{d x} ( φ ( x ^{2} + 2 y ( φ , x ) ) + ψ ( φ , x , y ( φ , x ) ) ( x - 2 y ( φ , x ) ( φ , x ) ^{2} ) )}{x ^{2}}

\frac{d}{d x} (φ (x^{2} + 2 y (φ, x)) + ψ (φ, x, y (φ, x)) (x - 2 y (φ, x) (φ, x)^{2})) = φ (2 x + 2 \frac{d}{d x} (y (φ, x))) + (\frac{d}{d x} (ψ (φ, x, y (φ, x))) (x - 2 y (φ, x) (φ, x)^{2}) + (1 - 2 (\frac{d}{d x} (y (φ, x)) (φ, x)^{2} + 2 (φ, x) (φ, 1) y (φ, x))) ψ (φ, x, y (φ, x)))

\frac{d x}{d x} = 1

= \frac{( φ ( 2 x + 2 \frac{d}{d x} ( y ( φ , x ) ) ) + ( \frac{d}{d x} ( ψ ( φ , x , y ( φ , x ) ) ) ( x - 2 y ( φ , x ) ( φ , x ) ^{2} ) + ( 1 - 2 ( \frac{d}{d x} ( y ( φ , x ) ) ( φ , x ) ^{2} + 2 ( φ , x ) ( φ , 1 ) y ( φ , x ) ) ) ψ ( φ , x , y ( φ , x ) ) ) ) x - 1 \cdot ( φ ( x ^{2} + 2 y ( φ , x ) ) + ψ ( φ , x , y ( φ , x ) ) ( x - 2 y ( φ , x ) ( φ , x ) ^{2} ) )}{x ^{2}}

\frac{e ^{- 2 π s}}{( s ^{2} + 1 ) \cdot ( s ^{2} - s + 4 )}

\frac{- 2 s ^{4} + 11 s ^{3} + 14 s ^{2} + s + 2}{s ^{2} \cdot ( 1 + 5 s - s ^{3} )}

(3 e^{x} + 2)^{2}

(10 e^{t} + 2) (t^{4} + 1)

\frac{( x + 1 ) ( x - 5 )}{( x + 5 ) ( x - 1 )}

What is the derivative of ((φ(x^2+2y(φ,x)+ψ(φ,x,y(φ,x))(x-2y(φ,x)^2)))/x) ?
The derivative of ((φ(x^2+2y(φ,x)+ψ(φ,x,y(φ,x))(x-2y(φ,x)^2)))/x) is
What is the first derivative of ((φ(x^2+2y(φ,x)+ψ(φ,x,y(φ,x))(x-2y(φ,x)^2)))/x) ?
The first derivative of ((φ(x^2+2y(φ,x)+ψ(φ,x,y(φ,x))(x-2y(φ,x)^2)))/x) is