What is the d/(dy)(ycos(y)) ?

The d/(dy)(ycos(y)) is cos(y)-ysin(y)

What is the first d/(dy)(ycos(y)) ?

The first d/(dy)(ycos(y)) is cos(y)-ysin(y)

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\frac{d}{d y} (y cos (y))

cos (y) - y sin (y)

Solution steps

\frac{d}{d y} (y cos (y))

Apply the Product Rule:

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

f = y, g = cos (y)

= \frac{d y}{d y} cos (y) + \frac{d}{d y} (cos (y)) y

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

\frac{d}{d y} (cos (y)) = - sin (y)

= 1 \cdot cos (y) + (- sin (y)) y

Simplify

= cos (y) - y sin (y)

f (x) = \frac{x ^{2}}{x - 6}

\int sin^{2} (\frac{1}{4} x) d x

x \cdot \frac{d y}{d x} + y = - 6 x y^{2}, y (1) = - 7

\frac{d}{d x} (e^{(e^{x^{2}})})

\int \frac{e \frac{1}{t} + 2}{t ^{2}} d t