What is (\partial)/(\partial y)(y(y+sin(x))) ?

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

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(\partial)/(\partial y)(y(y+sin(x)))

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

2 y + sin (x)

Solution steps

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

Treat

x

as a constant

Apply the Product Rule:

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

f = y, g = y + sin (x)

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

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

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

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

Simplify

1 \cdot (y + sin (x)) + 1 \cdot y : 2 y + sin (x)

= 2 y + sin (x)

\int_{0}^{0.6} \frac{x ^{2}}{9 - 25 x ^{2}} d x

2 x^{4} - 9 x^{2}

x y^{^{'}} + y = 3

t y^{^{'}} + (t + 1) y = t, y (ln (3)) = 1

x \to 0 lim (x^{4} (cos (\frac{2}{x})))

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