What is the d/(dy)(x^2ysin^3(xy^2)) ?

The d/(dy)(x^2ysin^3(xy^2)) is x^2(sin^3(xy^2)+6xy^2sin^2(xy^2)cos(xy^2))

What is the first d/(dy)(x^2ysin^3(xy^2)) ?

The first d/(dy)(x^2ysin^3(xy^2)) is x^2(sin^3(xy^2)+6xy^2sin^2(xy^2)cos(xy^2))

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\frac{d}{d y} (x^{2} y sin^{3} (x y^{2}))

x^{2} (sin^{3} (x y^{2}) + 6 x y^{2} sin^{2} (x y^{2}) cos (x y^{2}))

Solution steps

\frac{d}{d y} (x^{2} y sin^{3} (x y^{2}))

Treat

x

as a constant

Take the constant out:

(a \cdot f)^{'} = a \cdot f^{'}

= x^{2} \frac{d}{d y} (y sin^{3} (x y^{2}))

Apply the Product Rule:

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

f = y, g = sin^{3} (x y^{2})

= x^{2} (\frac{d y}{d y} sin^{3} (x y^{2}) + \frac{d}{d y} (sin^{3} (x y^{2})) y)

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

\frac{d}{d y} (sin^{3} (x y^{2})) = 6 x y sin^{2} (x y^{2}) cos (x y^{2})

= x^{2} (1 \cdot sin^{3} (x y^{2}) + 6 x y sin^{2} (x y^{2}) cos (x y^{2}) y)

Simplify

x^{2} (1 \cdot sin^{3} (x y^{2}) + 6 x y sin^{2} (x y^{2}) cos (x y^{2}) y) : x^{2} (sin^{3} (x y^{2}) + 6 x y^{2} sin^{2} (x y^{2}) cos (x y^{2}))

= x^{2} (sin^{3} (x y^{2}) + 6 x y^{2} sin^{2} (x y^{2}) cos (x y^{2}))

x \to 0 lim (arcsin (e^{x}))

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

\frac{d}{d x} (\frac{5 x ^{4}}{2 - x ^{2}})

x \to \infty lim (\frac{x ^{2} - 9}{2 x - 6})

y^{^{'}} - y tan (x) = cos (x)

What is the d/(dy)(x^2ysin^3(xy^2)) ?
The d/(dy)(x^2ysin^3(xy^2)) is x^2(sin^3(xy^2)+6xy^2sin^2(xy^2)cos(xy^2))
What is the first d/(dy)(x^2ysin^3(xy^2)) ?
The first d/(dy)(x^2ysin^3(xy^2)) is x^2(sin^3(xy^2)+6xy^2sin^2(xy^2)cos(xy^2))