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Calculus · Derivatives

Differentiate e^(-pi^2t)·sin(pi·x)

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

Step by step

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

    Start from the derivative to compute.

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

    Constant multiple rule: pull the constant out.

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

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

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

    Constant multiple rule: pull the constant out.

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

    d/dx of x is 1.

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