Solve any maths problem

Step by step

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

   Differentiate 2 times, one derivative at a time.

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

   Constant multiple rule: pull the constant out.

3. \frac{e^{- \frac{x^{2}}{4 t}} \frac{\partial}{\partial x} \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{d}{d x} x^{2}}{4 t^{\frac{3}{2}}}

   Constant multiple rule: pull the constant out.

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

   Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = 2.

6. f^{(1)}(x) = - \frac{x e^{- \frac{x^{2}}{4 t}}}{2 t^{\frac{3}{2}}}

   That is derivative number 1.

7. - \frac{\frac{\partial}{\partial x} x e^{- \frac{x^{2}}{4 t}}}{2 t^{\frac{3}{2}}}

   Constant multiple rule: pull the constant out.

8. - \frac{x \frac{\partial}{\partial x} e^{- \frac{x^{2}}{4 t}} + e^{- \frac{x^{2}}{4 t}} \frac{d}{d x} x}{2 t^{\frac{3}{2}}}

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

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

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

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

    d/dx of x is 1.

11. - \frac{e^{- \frac{x^{2}}{4 t}} - \frac{x e^{- \frac{x^{2}}{4 t}} \frac{d}{d x} x^{2}}{4 t}}{2 t^{\frac{3}{2}}}

    Constant multiple rule: pull the constant out.

12. - \frac{e^{- \frac{x^{2}}{4 t}} - \frac{x^{2} e^{- \frac{x^{2}}{4 t}}}{2 t}}{2 t^{\frac{3}{2}}}

    Power rule: (xⁿ)′ = n·xⁿ⁻¹ with n = 2.

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

    Simplify.

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

    That is derivative number 2.