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

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

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

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

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

Solution steps

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

Treat

c, t

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

- \frac{1}{( c ^{2} t ^{2} - x ^{2} ) ^{2}} \frac{\partial}{\partial x} (c^{2} t^{2} - x^{2})

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

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

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

Simplify

t (- \frac{1}{( c ^{2} t ^{2} - x ^{2} ) ^{2}} (- 2 x)) : \frac{2 t x}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}

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

f (x) = \frac{12}{x + 7}

\frac{\partial}{\partial x} (\frac{2 x - y}{5 x y + 6})

f (t) = arcsin (t)

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

\frac{\partial}{\partial x} (x^{6} y^{7} - x^{5} y^{6})

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