Question

Leibintz's theorem of x^3cosx

Answer 1

Answer:

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Step-by-step explanation:

Answer

I’ll offer an alternative approach — induction. The first step is to find some derivatives and look for a pattern. I’ll save you the leg-work; here they are:

f(x)=x2cos(x)f(x)=x2cos⁡(x)

f(1)(x)=−x2sin(x)+2xcos(x)f(1)(x)=−x2sin⁡(x)+2xcos⁡(x)

f(2)(x)=−x2cos(x)−4xsin(x)+2cos(x)f(2)(x)=−x2cos⁡(x)−4xsin⁡(x)+2cos⁡(x)

f(3)(x)=x2sin(x)−6xcos(x)−6sin(x)f(3)(x)=x2sin⁡(x)−6xcos⁡(x)−6sin⁡(x)

f(4)(x)=x2cos(x)+8xsin(x)−12cos(x)f(4)(x)=x2cos⁡(x)+8xsin⁡(x)−12cos⁡(x)

Hopefully by now you’re beginning to see a pattern: we might conjecture:

f(n)(x)=