Let , where and are real constants, not both zero. Define and choose an angle…

MATHEMATICS · Subjective · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Let g(x)=eaxsin⁡(bx)g(x)=e^{ax}\sin(bx), where aa and bb are real constants, not both zero. Define R=a2+b2R=\sqrt{a^{2}+b^{2}} and choose an angle θ\theta satisfying cos⁡θ=aR\cos\theta=\dfrac{a}{R} and sin⁡θ=bR\sin\theta=\dfrac{b}{R}. For a positive integer nn, derive a closed-form expression for g(n)(x)g^{(n)}(x) in terms of RR, θ\theta, nn, and xx.

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