What is (\partial)/(\partial y)((4sqrt(x))/(3y^2+1)) ?

The answer to (\partial)/(\partial y)((4sqrt(x))/(3y^2+1)) is -(24ysqrt(x))/((3y^2+1)^2)

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(\partial)/(\partial y)((4sqrt(x))/(3y^2+1))

\frac{\partial}{\partial y} (\frac{4 x}{3 y ^{2} + 1})

- \frac{24 y x}{( 3 y ^{2} + 1 ) ^{2}}

Solution steps

\frac{\partial}{\partial y} (\frac{4 x}{3 y ^{2} + 1})

Treat

x

as a constant

Take the constant out:

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

= 4 x \frac{\partial}{\partial y} (\frac{1}{3 y ^{2} + 1})

Apply exponent rule:

\frac{1}{a} = a^{- 1}

= 4 x \frac{\partial}{\partial y} ((3 y^{2} + 1)^{- 1})

Apply the chain rule:

- \frac{1}{( 3 y ^{2} + 1 ) ^{2}} \frac{\partial}{\partial y} (3 y^{2} + 1)

= - \frac{1}{( 3 y ^{2} + 1 ) ^{2}} \frac{\partial}{\partial y} (3 y^{2} + 1)

\frac{\partial}{\partial y} (3 y^{2} + 1) = 6 y

= 4 x (- \frac{1}{( 3 y ^{2} + 1 ) ^{2}} \cdot 6 y)

Simplify

4 x (- \frac{1}{( 3 y ^{2} + 1 ) ^{2}} \cdot 6 y) : - \frac{24 y x}{( 3 y ^{2} + 1 ) ^{2}}

= - \frac{24 y x}{( 3 y ^{2} + 1 ) ^{2}}

x \to 2 lim (3 + x)

\frac{d}{d x} (\frac{e ^{x}}{7 x ^{2}})

sin (x), 5 x, \frac{π}{2}, π

x \to - 2 lim (x + 3)

f (x) = (x^{2} - 5) e^{x}

What is (\partial)/(\partial y)((4sqrt(x))/(3y^2+1)) ?
The answer to (\partial)/(\partial y)((4sqrt(x))/(3y^2+1)) is -(24ysqrt(x))/((3y^2+1)^2)