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Calculus · Derivatives

Differentiate e^(-pi^2t)·sin(pi·x)

\frac{d^2}{dx^2}\left[e^{- \pi^{2} t} \sin{\left(\pi x \right)}\right]

Step by step

  1. \frac{d}{dx}\left[e^{- \pi^{2} t} \sin{\left(\pi x \right)}\right]

    Differentiate 2 times, one derivative at a time.

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

    Constant multiple rule: pull the constant out.

  3. e^{- \pi^{2} t} \cos{\left(\pi x \right)} \frac{d}{d x} \pi x

    Chain rule: (sin u)′ = cos u, times u′ where u = \pi x.

  4. \pi e^{- \pi^{2} t} \cos{\left(\pi x \right)} \frac{d}{d x} x

    Constant multiple rule: pull the constant out.

  5. \pi e^{- \pi^{2} t} \cos{\left(\pi x \right)}

    d/dx of x is 1.

  6. f^{(1)}(x) = \pi e^{- \pi^{2} t} \cos{\left(\pi x \right)}

    That is derivative number 1.

  7. \pi e^{- \pi^{2} t} \frac{d}{d x} \cos{\left(\pi x \right)}

    Constant multiple rule: pull the constant out.

  8. - \pi e^{- \pi^{2} t} \sin{\left(\pi x \right)} \frac{d}{d x} \pi x

    Chain rule: (cos u)′ = −sin u, times u′ where u = \pi x.

  9. - \pi^{2} e^{- \pi^{2} t} \sin{\left(\pi x \right)} \frac{d}{d x} x

    Constant multiple rule: pull the constant out.

  10. - \pi^{2} e^{- \pi^{2} t} \sin{\left(\pi x \right)}

    d/dx of x is 1.

  11. f^{(2)}(x) = - \pi^{2} e^{- \pi^{2} t} \sin{\left(\pi x \right)}

    That is derivative number 2.

Reveal the answer
f''(x) = - \pi^{2} e^{- \pi^{2} t} \sin{\left(\pi x \right)}