Define for all real .

MATHEMATICS · Multi-part · AP · Calculus AB · Differentiation: Composite, Implicit, and Inverse Functions · Differentiation: Composite, Implicit, and Inverse Functions

Problem

Define f(x)=e5xsin⁡xf(x)=e^{5x}\sin x for all real xx. (a) Derive an explicit expression for the nthn^{\text{th}} derivative f(n)(x)f^{(n)}(x) that involves only e5xe^{5x} multiplied by either sin⁡x\sin x or cos⁡x\cos x. Your formula should clearly indicate how the trigonometric factor and its coefficient depend on nn. Provide justification for the pattern you state. (b) Use your result from part (a) to evaluate f(6) ⁣(π2)f^{(6)}\!\left(\dfrac{\pi}{2}\right). Give exact values, not decimals.

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