What is (\partial)/(\partial x)((e^x(y-x))dx) ?

The answer to (\partial)/(\partial x)((e^x(y-x))dx) is d(e^x(y-x)x+(-2x+y)e^x)

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(\partial)/(\partial x)((e^x(y-x))dx)

\frac{\partial}{\partial x} ((e^{x} (y - x)) d x)

d (e^{x} (y - x) x + (- 2 x + y) e^{x})

Solution steps

\frac{\partial}{\partial x} ((e^{x} (y - x)) d x)

Treat

y, d

as constants

Take the constant out:

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

= d \frac{\partial}{\partial x} (e^{x} (y - x) x)

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{x}, g = (y - x) x

= d (\frac{\partial}{\partial x} (e^{x}) (y - x) x + \frac{\partial}{\partial x} ((y - x) x) e^{x})

\frac{\partial}{\partial x} (e^{x}) = e^{x}

\frac{\partial}{\partial x} ((y - x) x) = - 2 x + y

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

\int \frac{x}{2 x ^{2} + 4} d x

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

\frac{d}{d x} (2 sin (x - 2))

\frac{d}{d x} (9 - x^{\frac{2}{3}})

(cos^{3} (x))^{^{'}}

What is (\partial)/(\partial x)((e^x(y-x))dx) ?
The answer to (\partial)/(\partial x)((e^x(y-x))dx) is d(e^x(y-x)x+(-2x+y)e^x)