What is (\partial}{\partial x}(\frac{(a{R}(x)+b+c{R}(x)+de+{f}(x)+{g}(x)e))/x) ?

The answer to (\partial}{\partial x}(\frac{(a{R}(x)+b+c{R}(x)+de+{f}(x)+{g}(x)e))/x) is

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(\partial}{\partial x}(\frac{(a{R}(x)+b+c{R}(x)+de+{f}(x)+{g}(x)e))/x)

\frac{\partial}{\partial x} (\frac{( a R ( x ) + b + c R ( x ) + d e + f ( x ) + g ( x ) e )}{x})

\frac{a x \frac{\partial}{\partial x} ( R ( x ) ) + c x \frac{\partial}{\partial x} ( R ( x ) ) + x \frac{\partial}{\partial x} ( f ( x ) ) + e x \frac{\partial}{\partial x} ( g ( x ) ) - a R ( x ) - b - c R ( x ) - e d - f ( x ) - e g ( x )}{x ^{2}}

Solution steps

\frac{\partial}{\partial x} (\frac{( a R ( x ) + b + c R ( x ) + d e + f ( x ) + g ( x ) e )}{x})

Apply the Quotient Rule:

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

= \frac{\frac{\partial}{\partial x} ( a R ( x ) + b + c R ( x ) + d e + f ( x ) + g ( x ) e ) x - \frac{\partial}{\partial x} ( x ) ( a R ( x ) + b + c R ( x ) + d e + f ( x ) + g ( x ) e )}{x ^{2}}

\frac{\partial}{\partial x} (a R (x) + b + c R (x) + d e + f (x) + g (x) e) = a \frac{\partial}{\partial x} (R (x)) + c \frac{\partial}{\partial x} (R (x)) + \frac{\partial}{\partial x} (f (x)) + e \frac{\partial}{\partial x} (g (x))

\frac{\partial}{\partial x} (x) = 1

= \frac{( a \frac{\partial}{\partial x} ( R ( x ) ) + c \frac{\partial}{\partial x} ( R ( x ) ) + \frac{\partial}{\partial x} ( f ( x ) ) + e \frac{\partial}{\partial x} ( g ( x ) ) ) x - 1 \cdot ( a R ( x ) + b + c R ( x ) + d e + f ( x ) + g ( x ) e )}{x ^{2}}

Simplify

\frac{( a \frac{\partial}{\partial x} ( R ( x ) ) + c \frac{\partial}{\partial x} ( R ( x ) ) + \frac{\partial}{\partial x} ( f ( x ) ) + e \frac{\partial}{\partial x} ( g ( x ) ) ) x - 1 \cdot ( a R ( x ) + b + c R ( x ) + d e + f ( x ) + g ( x ) e )}{x ^{2}} : \frac{a x \frac{\partial}{\partial x} ( R ( x ) ) + c x \frac{\partial}{\partial x} ( R ( x ) ) + x \frac{\partial}{\partial x} ( f ( x ) ) + e x \frac{\partial}{\partial x} ( g ( x ) ) - a R ( x ) - b - c R ( x ) - e d - f ( x ) - e g ( x )}{x ^{2}}

= \frac{a x \frac{\partial}{\partial x} ( R ( x ) ) + c x \frac{\partial}{\partial x} ( R ( x ) ) + x \frac{\partial}{\partial x} ( f ( x ) ) + e x \frac{\partial}{\partial x} ( g ( x ) ) - a R ( x ) - b - c R ( x ) - e d - f ( x ) - e g ( x )}{x ^{2}}

y ln (x) y^{^{'}} = (\frac{( y + 1 )}{x ^{(\frac{1}{2})}})^{2}

\int e^{5 x} d x

\int 12 x^{5} d x

\frac{\partial}{\partial t} (- t e^{t} sin (x))

\int_{0}^{5} ∣ 2 x - 5 ∣ d x

What is (\partial}{\partial x}(\frac{(a{R}(x)+b+c{R}(x)+de+{f}(x)+{g}(x)e))/x) ?
The answer to (\partial}{\partial x}(\frac{(a{R}(x)+b+c{R}(x)+de+{f}(x)+{g}(x)e))/x) is