What is (\partial)/(\partial x)(xsin(x^2-y^2)) ?

The answer to (\partial)/(\partial x)(xsin(x^2-y^2)) is sin(x^2-y^2)+2x^2cos(x^2-y^2)

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(\partial)/(\partial x)(xsin(x^2-y^2))

\frac{\partial}{\partial x} (x sin (x^{2} - y^{2}))

sin (x^{2} - y^{2}) + 2 x^{2} cos (x^{2} - y^{2})

Solution steps

\frac{\partial}{\partial x} (x sin (x^{2} - y^{2}))

Treat

y

as a constant

Apply the Product Rule:

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

f = x, g = sin (x^{2} - y^{2})

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

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

\frac{\partial}{\partial x} (sin (x^{2} - y^{2})) = cos (x^{2} - y^{2}) \cdot 2 x

= 1 \cdot sin (x^{2} - y^{2}) + cos (x^{2} - y^{2}) \cdot 2 xx

Simplify

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

= sin (x^{2} - y^{2}) + 2 x^{2} cos (x^{2} - y^{2})

(x + y e^{\frac{y}{x}}) d x - (x e^{\frac{y}{x}}) d y = 0

\frac{d}{d x} (\frac{- x ^{2} - 5}{( x ^{2} - 5 ) ^{2}})

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

\frac{d}{d x} (- 7 csc (x) + \frac{1}{x ^{5}})

arcsin (\frac{x}{a})

What is (\partial)/(\partial x)(xsin(x^2-y^2)) ?
The answer to (\partial)/(\partial x)(xsin(x^2-y^2)) is sin(x^2-y^2)+2x^2cos(x^2-y^2)