What is the d/(dy)((x(y))((y^6)/6+1/(16y^4))^2) ?

The d/(dy)((x(y))((y^6)/6+1/(16y^4))^2) is (x(832y^{20}+144y^{10}-63))/(2304y^8)

What is the first d/(dy)((x(y))((y^6)/6+1/(16y^4))^2) ?

The first d/(dy)((x(y))((y^6)/6+1/(16y^4))^2) is (x(832y^{20}+144y^{10}-63))/(2304y^8)

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\frac{d}{d y} ((x (y)) (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2})

\frac{x ( 832 y ^{20} + 144 y ^{10} - 63 )}{2304 y ^{8}}

Solution steps

\frac{d}{d y} ((x (y)) (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2})

Treat

x

as a constant

Take the constant out:

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

= x \frac{d}{d y} (y (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2})

Apply the Product Rule:

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

f = y, g = (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2}

= x (\frac{d y}{d y} (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2} + \frac{d}{d y} ((\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2}) y)

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

\frac{d}{d y} ((\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2}) = 2 (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}}) (y^{5} - \frac{1}{4 y ^{5}})

= x (1 \cdot (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2} + 2 (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}}) (y^{5} - \frac{1}{4 y ^{5}}) y)

Simplify

x (1 \cdot (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}})^{2} + 2 (\frac{y ^{6}}{6} + \frac{1}{16 y ^{4}}) (y^{5} - \frac{1}{4 y ^{5}}) y) : \frac{x ( 832 y ^{20} + 144 y ^{10} - 63 )}{2304 y ^{8}}

= \frac{x ( 832 y ^{20} + 144 y ^{10} - 63 )}{2304 y ^{8}}

\frac{2 ( 2 i z ^{2} - 15 z ^{2} + 15 i z + 16 z - 14 i )}{( z - i ) ^{5}}

(\frac{2 x + 1}{4 x}) \cdot (\frac{- 16}{x ( 1 + x ) ( x - 4 ) ^{2}})

2 y (x y^{'} (x) - y)

π x^{2} \cdot (9 - x) - 90

(2 x + e^{x}) (x^{3} + 2 e^{2 x})

What is the d/(dy)((x(y))((y^6)/6+1/(16y^4))^2) ?
The d/(dy)((x(y))((y^6)/6+1/(16y^4))^2) is (x(832y^{20}+144y^{10}-63))/(2304y^8)
What is the first d/(dy)((x(y))((y^6)/6+1/(16y^4))^2) ?
The first d/(dy)((x(y))((y^6)/6+1/(16y^4))^2) is (x(832y^{20}+144y^{10}-63))/(2304y^8)