Solve any maths problem

Step by step

1. \frac{d}{dt}\left[e^{- \left(t + x\right)^{2}}\right]

   Differentiate 2 times, one derivative at a time.

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

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

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

   Constant multiple rule: pull the constant out.

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

   Chain rule + power rule: (uⁿ)′ = n·uⁿ⁻¹·u′ with u = t + x, n = 2.

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

   Sum rule: differentiate term by term.

6. - \left(2 t + 2 x\right) \left(\frac{d}{d t} x + 1\right) e^{- \left(t + x\right)^{2}}

   d/dx of x is 1.

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

   The derivative of a constant is 0.

8. f^{(1)}(t) = - \left(2 t + 2 x\right) e^{- \left(t + x\right)^{2}}

   That is derivative number 1.

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

   Constant multiple rule: pull the constant out.

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

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

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

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

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

    Sum rule: differentiate term by term.

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

    Constant multiple rule: pull the constant out.

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

    d/dx of x is 1.

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

    The derivative of a constant is 0.

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

    Constant multiple rule: pull the constant out.

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

    Chain rule + power rule: (uⁿ)′ = n·uⁿ⁻¹·u′ with u = t + x, n = 2.

18. \left(2 t + 2 x\right)^{2} \left(\frac{d}{d t} t + \frac{d}{d t} x\right) e^{- \left(t + x\right)^{2}} - 2 e^{- \left(t + x\right)^{2}}

    Sum rule: differentiate term by term.

19. \left(2 t + 2 x\right)^{2} \left(\frac{d}{d t} x + 1\right) e^{- \left(t + x\right)^{2}} - 2 e^{- \left(t + x\right)^{2}}

    d/dx of x is 1.

20. \left(2 t + 2 x\right)^{2} e^{- \left(t + x\right)^{2}} - 2 e^{- \left(t + x\right)^{2}}

    The derivative of a constant is 0.

21. 2 \left(2 \left(t + x\right)^{2} - 1\right) e^{- \left(t + x\right)^{2}}

    Simplify.

22. f^{(2)}(t) = 2 \left(2 \left(t + x\right)^{2} - 1\right) e^{- \left(t + x\right)^{2}}

    That is derivative number 2.