What is (\partial)/(\partial x)((rx(K(r,x)-x))/(K(r,x)+ax)) ?

The answer to (\partial)/(\partial x)((rx(K(r,x)-x))/(K(r,x)+ax)) is

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(\partial)/(\partial x)((rx(K(r,x)-x))/(K(r,x)+ax))

\frac{\partial}{\partial x} (\frac{r x ( K ( r , x ) - x )}{K ( r , x ) + a x})

r \frac{( 1 \cdot ( K ( r , x ) - x ) + ( \frac{\partial}{\partial x} ( K ( r , x ) ) - 1 ) x ) ( K ( r , x ) + a x ) - ( \frac{\partial}{\partial x} ( K ( r , x ) ) + a ) x ( K ( r , x ) - x )}{( K ( r , x ) + a x ) ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{r x ( K ( r , x ) - x )}{K ( r , x ) + a x})

Treat

a, r

as a constant

Take the constant out:

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

= r \frac{\partial}{\partial x} (\frac{x ( K ( r , x ) - x )}{K ( r , x ) + a x})

Apply the Quotient Rule:

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

= r \frac{\frac{\partial}{\partial x} ( x ( K ( r , x ) - x ) ) ( K ( r , x ) + a x ) - \frac{\partial}{\partial x} ( K ( r , x ) + a x ) x ( K ( r , x ) - x )}{( K ( r , x ) + a x ) ^{2}}

\frac{\partial}{\partial x} (x (K (r, x) - x)) = 1 \cdot (K (r, x) - x) + (\frac{\partial}{\partial x} (K (r, x)) - 1) x

\frac{\partial}{\partial x} (K (r, x) + a x) = \frac{\partial}{\partial x} (K (r, x)) + a

= r \frac{( 1 \cdot ( K ( r , x ) - x ) + ( \frac{\partial}{\partial x} ( K ( r , x ) ) - 1 ) x ) ( K ( r , x ) + a x ) - ( \frac{\partial}{\partial x} ( K ( r , x ) ) + a ) x ( K ( r , x ) - x )}{( K ( r , x ) + a x ) ^{2}}

What is (\partial)/(\partial x)((rx(K(r,x)-x))/(K(r,x)+ax)) ?
The answer to (\partial)/(\partial x)((rx(K(r,x)-x))/(K(r,x)+ax)) is