What is (\partial)/(\partial t)((rs)/(s^2+t^2)) ?

The answer to (\partial)/(\partial t)((rs)/(s^2+t^2)) is -(2rst)/((s^2+t^2)^2)

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(\partial)/(\partial t)((rs)/(s^2+t^2))

\frac{\partial}{\partial t} (\frac{rs}{s ^{2} + t ^{2}})

- \frac{2 rs t}{( s ^{2} + t ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial t} (\frac{rs}{s ^{2} + t ^{2}})

Treat

r, s

as constants

Take the constant out:

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

= rs \frac{\partial}{\partial t} (\frac{1}{s ^{2} + t ^{2}})

Apply exponent rule:

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

= rs \frac{\partial}{\partial t} ((s^{2} + t^{2})^{- 1})

Apply the chain rule:

- \frac{1}{( s ^{2} + t ^{2} ) ^{2}} \frac{\partial}{\partial t} (s^{2} + t^{2})

= - \frac{1}{( s ^{2} + t ^{2} ) ^{2}} \frac{\partial}{\partial t} (s^{2} + t^{2})

\frac{\partial}{\partial t} (s^{2} + t^{2}) = 2 t

= rs (- \frac{1}{( s ^{2} + t ^{2} ) ^{2}} \cdot 2 t)

Simplify

rs (- \frac{1}{( s ^{2} + t ^{2} ) ^{2}} \cdot 2 t) : - \frac{2 rs t}{( s ^{2} + t ^{2} ) ^{2}}

= - \frac{2 rs t}{( s ^{2} + t ^{2} ) ^{2}}

\frac{d}{d x} ((8 x^{2} + 3) (3 - x^{2} + 7 x))

\frac{e ^{- s}}{s - 2}

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

f (x) = 7 x + 11, at x = 2

\frac{d y}{d x} = \frac{cos ( 8 x )}{e ^{8 y}}

What is (\partial)/(\partial t)((rs)/(s^2+t^2)) ?
The answer to (\partial)/(\partial t)((rs)/(s^2+t^2)) is -(2rst)/((s^2+t^2)^2)