What is (\partial)/(\partial t)(2/((x-3t)^2+1)) ?

The answer to (\partial)/(\partial t)(2/((x-3t)^2+1)) is (12(-3t+x))/(((x-3t)^2+1)^2)

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(\partial)/(\partial t)(2/((x-3t)^2+1))

\frac{\partial}{\partial t} (\frac{2}{( x - 3 t ) ^{2} + 1})

\frac{12 ( - 3 t + x )}{( ( x - 3 t ) ^{2} + 1 ) ^{2}}

Solution steps

\frac{\partial}{\partial t} (\frac{2}{( x - 3 t ) ^{2} + 1})

Treat

x

as a constant

Take the constant out:

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

= 2 \frac{\partial}{\partial t} (\frac{1}{( x - 3 t ) ^{2} + 1})

Apply exponent rule:

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

= 2 \frac{\partial}{\partial t} (((x - 3 t)^{2} + 1)^{- 1})

Apply the chain rule:

- \frac{1}{( ( x - 3 t ) ^{2} + 1 ) ^{2}} \frac{\partial}{\partial t} ((x - 3 t)^{2} + 1)

= - \frac{1}{( ( x - 3 t ) ^{2} + 1 ) ^{2}} \frac{\partial}{\partial t} ((x - 3 t)^{2} + 1)

\frac{\partial}{\partial t} ((x - 3 t)^{2} + 1) = - 6 (- 3 t + x)

= 2 - \frac{1}{( ( x - 3 t ) ^{2} + 1 ) ^{2}} (- 6 (- 3 t + x))

Simplify

2 - \frac{1}{( ( x - 3 t ) ^{2} + 1 ) ^{2}} (- 6 (- 3 t + x)) : \frac{12 ( - 3 t + x )}{( ( x - 3 t ) ^{2} + 1 ) ^{2}}

= \frac{12 ( - 3 t + x )}{( ( x - 3 t ) ^{2} + 1 ) ^{2}}

What is (\partial)/(\partial t)(2/((x-3t)^2+1)) ?
The answer to (\partial)/(\partial t)(2/((x-3t)^2+1)) is (12(-3t+x))/(((x-3t)^2+1)^2)