What is (\partial)/(\partial y)(xsqrt(y^2+1)) ?

The answer to (\partial)/(\partial y)(xsqrt(y^2+1)) is (xy)/(sqrt(y^2+1))

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(\partial)/(\partial y)(xsqrt(y^2+1))

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

\frac{x y}{y ^{2} + 1}

Solution steps

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

Treat

x

as a constant

Take the constant out:

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

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

Apply the chain rule:

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

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

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

= x \frac{1}{2 y ^{2} + 1} \cdot 2 y

Simplify

x \frac{1}{2 y ^{2} + 1} \cdot 2 y : \frac{x y}{y ^{2} + 1}

= \frac{x y}{y ^{2} + 1}

y^{^{''}} - 43 y^{^{'}} + 12 y = 0

\frac{d}{d x} (\frac{1}{3} ln (e^{x} + 1))

\frac{d}{d x} (- \frac{2 x}{1 - x ^{2}})

f (x) = - x^{2} sin (x) + 4 x cos (x) + 2 sin (x)

\frac{d}{d x} (\frac{3}{x ^{4}})

What is (\partial)/(\partial y)(xsqrt(y^2+1)) ?
The answer to (\partial)/(\partial y)(xsqrt(y^2+1)) is (xy)/(sqrt(y^2+1))