What is (\partial)/(\partial t)(t/(c^2t^2-x^2)) ?

The answer to (\partial)/(\partial t)(t/(c^2t^2-x^2)) is (-c^2t^2-x^2)/((c^2t^2-x^2)^2)

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(\partial)/(\partial t)(t/(c^2t^2-x^2))

\frac{\partial}{\partial t} (\frac{t}{c ^{2} t ^{2} - x ^{2}})

\frac{- c ^{2} t ^{2} - x ^{2}}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}

Solution steps

\frac{\partial}{\partial t} (\frac{t}{c ^{2} t ^{2} - x ^{2}})

Treat

c, x

as a constant

Apply the Quotient Rule:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{\partial}{\partial t} ( t ) ( c ^{2} t ^{2} - x ^{2} ) - \frac{\partial}{\partial t} ( c ^{2} t ^{2} - x ^{2} ) t}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}

\frac{\partial}{\partial t} (t) = 1

\frac{\partial}{\partial t} (c^{2} t^{2} - x^{2}) = 2 c^{2} t

= \frac{1 \cdot ( c ^{2} t ^{2} - x ^{2} ) - 2 c ^{2} tt}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}

1 \cdot (c^{2} t^{2} - x^{2}) - 2 c^{2} tt = - c^{2} t^{2} - x^{2}

= \frac{- c ^{2} t ^{2} - x ^{2}}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}

y^{^{''}} + 12 y^{^{'}} + 36 = 0

\frac{\partial}{\partial x} (y e^{x} + x ln (z))

t y^{^{'}} = 4 t^{2} - 2 y

y^{^{''}} x = y^{^{'}}

\frac{\partial}{\partial u} (u v^{2} + w^{3})

What is (\partial)/(\partial t)(t/(c^2t^2-x^2)) ?
The answer to (\partial)/(\partial t)(t/(c^2t^2-x^2)) is (-c^2t^2-x^2)/((c^2t^2-x^2)^2)