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x(n)=z cosh(alpha n)u(n)

question: x(n)=z extcosh(extalpha n)u(n)x(n)=z \, ext{cosh}( ext{alpha} \, n)u(n)

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Find derivatives using the quotient rule Iright chevron icon

Full solution

Q. question: x(n)=z extcosh(extalpha n)u(n)x(n)=z \, ext{cosh}( ext{alpha} \, n)u(n)
  1. Identify Components: Identify the components. zz is a constant, extcosh(αn) ext{cosh}(\alpha n) is a hyperbolic cosine function, and u(n)u(n) is the unit step function.
  2. Derivative Calculation: Derivative of z cosh⁡(α n)z \, \cosh(\alpha \, n) with respect to nn is z α sinh⁡(α n)z \, \alpha \, \sinh(\alpha \, n).
  3. Unit Step Function Derivative: Derivative of u(n)u(n) with respect to nn is δ(n)\delta(n), the Dirac delta function.
  4. Apply Product Rule: Apply the product rule: ddn[zcosh⁡(αn)u(n)]=z⋅α⋅sinh⁡(αn)⋅u(n)+z⋅cosh⁡(αn)⋅δ(n)\frac{d}{dn} [z \cosh(\alpha n)u(n)] = z \cdot \alpha \cdot \sinh(\alpha n) \cdot u(n) + z \cdot \cosh(\alpha n) \cdot \delta(n).

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