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

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

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

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

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

Solution steps

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

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

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

\int π sin (x) d x

x \to 6 + lim (x - 6)

\frac{d}{d x} (\frac{x ^{2}}{x})

x \to 1.9 lim (- x^{2} + 5 x - 2)

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