What is the d/(dy)((x)(x^2+x)) ?

The d/(dy)((x)(x^2+x)) is 3y^2+2y

What is the first d/(dy)((x)(x^2+x)) ?

The first d/(dy)((x)(x^2+x)) is 3y^2+2y

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\frac{d}{d y} ((y) (y^{2} + y))

3 y^{2} + 2 y

Solution steps

\frac{d}{d y} ((y) (y^{2} + y))

Apply the Product Rule:

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

f = y, g = y^{2} + y

= \frac{d y}{d y} (y^{2} + y) + \frac{d}{d y} (y^{2} + y) y

\frac{d y}{d y} = 1

\frac{d}{d y} (y^{2} + y) = 2 y + 1

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

Simplify

1 \cdot (y^{2} + y) + (2 y + 1) y : 3 y^{2} + 2 y

= 3 y^{2} + 2 y

(sin (y) - y sin (x)) d y

e^{2 x} (y - 2)

- 2 (4 x ln (x) + 2 x)

\frac{3 y ( 4 y + 1 )}{7 ( 4 y - 1 )}

\frac{121 ( 3 π - 8 )}{8} + 121 π