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

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

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

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

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

Solution steps

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

Treat

a, t

as a constant

Take the constant out:

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

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

Apply exponent rule:

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

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

Apply the chain rule:

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

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

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

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

Simplify

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

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

\frac{e ^{x} + 1}{e ^{x} - 1}

\frac{d y}{d x} = 2 x y^{2} + 128 x

x \to - 6 lim (12 x + 1)

\frac{d y}{d x} = \frac{1}{2} (y - 1)^{2}, y (0) = 2

\int \frac{x ^{3}}{x ^{2} + 4} d x

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