What is (\partial)/(\partial x)(-(v(x,t))/(b(x,t)(x-v(x,t)t))) ?

The answer to (\partial)/(\partial x)(-(v(x,t))/(b(x,t)(x-v(x,t)t))) is

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(\partial)/(\partial x)(-(v(x,t))/(b(x,t)(x-v(x,t)t)))

\frac{\partial}{\partial x} (- \frac{v ( x , t )}{b ( x , t ) ( x - v ( x , t ) t )})

- \frac{\frac{\partial}{\partial x} ( v ( x , t ) ) b ( x , t ) ( x - v ( x , t ) t ) - ( \frac{\partial}{\partial x} ( b ( x , t ) ) ( x - v ( x , t ) t ) + ( 1 - t \frac{\partial}{\partial x} ( v ( x , t ) ) ) b ( x , t ) ) v ( x , t )}{( b ( x , t ) ( x - v ( x , t ) t ) ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (- \frac{v ( x , t )}{b ( x , t ) ( x - v ( x , t ) t )})

Treat

t

as a constant

Take the constant out:

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

= - \frac{\partial}{\partial x} (\frac{v ( x , t )}{b ( x , t ) ( x - v ( x , t ) t )})

Apply the Quotient Rule:

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

= - \frac{\frac{\partial}{\partial x} ( v ( x , t ) ) b ( x , t ) ( x - v ( x , t ) t ) - \frac{\partial}{\partial x} ( b ( x , t ) ( x - v ( x , t ) t ) ) v ( x , t )}{( b ( x , t ) ( x - v ( x , t ) t ) ) ^{2}}

\frac{\partial}{\partial x} (b (x, t) (x - v (x, t) t)) = \frac{\partial}{\partial x} (b (x, t)) (x - v (x, t) t) + (1 - t \frac{\partial}{\partial x} (v (x, t))) b (x, t)

= - \frac{\frac{\partial}{\partial x} ( v ( x , t ) ) b ( x , t ) ( x - v ( x , t ) t ) - ( \frac{\partial}{\partial x} ( b ( x , t ) ) ( x - v ( x , t ) t ) + ( 1 - t \frac{\partial}{\partial x} ( v ( x , t ) ) ) b ( x , t ) ) v ( x , t )}{( b ( x , t ) ( x - v ( x , t ) t ) ) ^{2}}

\frac{d y}{y} = d x

\frac{1}{x}

\frac{1 - x e ^{x}}{x + e ^{x}}

(y^{2} - x^{3} - 2 x) d x + 2 x y d y = 0

x \to - 2 lim (\frac{10 x}{x ^{2}})

What is (\partial)/(\partial x)(-(v(x,t))/(b(x,t)(x-v(x,t)t))) ?
The answer to (\partial)/(\partial x)(-(v(x,t))/(b(x,t)(x-v(x,t)t))) is