Thursday, November 6, 2014

Single Variable Calculus, Chapter 3, Review Exercises, Section Review Exercises, Problem 44

Determine f(n)(x) if f(x)=1(2x)
Rewrite f(x) to f(x)=(2x)1, then solve for the 1st derivative.
We have,



f(x)=ddx(2x)1f(x)=1(2x)2ddx(2x)f(x)=1(2x)2(1)f(x)=(2x)2

Solving for the 2nd derivative




f(x)=ddx(2x)2f(x)=2(2x)3ddx(2x)f(x)=2(2x)3(1)f(x)=2(2x)3


Solving for the 3rd derivative



f(x)=2ddx(2x)3f(x)=(2)(3)(2x)4ddx(2x)f(x)=6(2x)4(1)f(x)=6(2x)4


By solving the first, second and third derivative of the function. We get the pattern,
f(n)=n!(2x)(1n)

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