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1. \frac{d}{dt}\left[\frac{e^{- \frac{x^{2}}{4 t}}}{\sqrt{t}}\right]

   Start from the derivative to compute.

2. e^{- \frac{x^{2}}{4 t}} \frac{d}{d t} \frac{1}{\sqrt{t}} + \frac{\frac{\partial}{\partial t} e^{- \frac{x^{2}}{4 t}}}{\sqrt{t}}

   Product rule with u = \frac{1}{\sqrt{t}} and v = e^{- \frac{x^{2}}{4 t}}: (uv)′ = u′v + uv′.

3. e^{- \frac{x^{2}}{4 t}} \frac{d}{d t} \frac{1}{\sqrt{t}} + \frac{e^{- \frac{x^{2}}{4 t}} \frac{\partial}{\partial t} \left(- \frac{x^{2}}{4 t}\right)}{\sqrt{t}}

   Chain rule: (eᵘ)′ = eᵘ, times u′ where u = - \frac{x^{2}}{4 t}.

4. \frac{e^{- \frac{x^{2}}{4 t}} \frac{\partial}{\partial t} \left(- \frac{x^{2}}{4 t}\right)}{\sqrt{t}} - \frac{e^{- \frac{x^{2}}{4 t}}}{2 t^{\frac{3}{2}}}

   Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = - \frac{1}{2}.

5. - \frac{x^{2} e^{- \frac{x^{2}}{4 t}} \frac{d}{d t} \frac{1}{t}}{4 \sqrt{t}} - \frac{e^{- \frac{x^{2}}{4 t}}}{2 t^{\frac{3}{2}}}

   Constant multiple rule: pull the constant out.

6. - \frac{e^{- \frac{x^{2}}{4 t}}}{2 t^{\frac{3}{2}}} + \frac{x^{2} e^{- \frac{x^{2}}{4 t}}}{4 t^{\frac{5}{2}}}

   Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = -1.

7. \frac{\left(- 2 t + x^{2}\right) e^{- \frac{x^{2}}{4 t}}}{4 t^{\frac{5}{2}}}

   Simplify.