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

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

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

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

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

Solution steps

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

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

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

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

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

Simplify

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

\frac{d}{d x} (\frac{5 x}{x - 1})

\int_{1}^{6} t ln (t) d t

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

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

5 sin^{2} (x)

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