What is (\partial)/(\partial x)((x-y)(1-xy)) ?

The answer to (\partial)/(\partial x)((x-y)(1-xy)) is 1-2xy+y^2

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(\partial)/(\partial x)((x-y)(1-xy))

\frac{\partial}{\partial x} ((x - y) (1 - x y))

1 - 2 x y + y^{2}

Solution steps

\frac{\partial}{\partial x} ((x - y) (1 - x y))

Treat

y

as a constant

Apply the Product Rule:

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

f = x - y, g = 1 - x y

= \frac{\partial}{\partial x} (x - y) (1 - x y) + \frac{\partial}{\partial x} (1 - x y) (x - y)

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

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

= 1 \cdot (1 - x y) + (- y) (x - y)

Simplify

1 \cdot (1 - x y) + (- y) (x - y) : 1 - 2 x y + y^{2}

= 1 - 2 x y + y^{2}

\int_{0}^{1} \frac{1}{x ^{2} + 1} d x

\int \frac{2}{2 - x ^{2}} d x

\int \frac{sin ( x )}{cos ^{4} ( x )} d x

\frac{5 x}{( 1 + x ^{2} )}

What is (\partial)/(\partial x)((x-y)(1-xy)) ?
The answer to (\partial)/(\partial x)((x-y)(1-xy)) is 1-2xy+y^2