Question

Find a formula for f^(101)(x) if f(x) = 1/(4x - 1).

Answer 1

Given,

f(x) = 1/(4x - 1).

To find,

a formula for f^(101)(x)

Let us consider,

$f^0(x)=\dfrac{1}{4x-1}$

$f^1(x)=-\dfrac{4}{\left(4x-1\right)^2}$

$f^2(x)=\dfrac{32}{\left(4x-1\right)^3}$

$f^3 (x) = -\dfrac{384}{\left(4x-1\right)^4}$

$f^4 (x) = \dfrac{6144}{\left(4x-1\right)^5}$

so on, upto f^(101), therefore, we have positive values for even power of f(x) and negative values for odd power of f(x).

as f^(101) is odd, so we will have negative values, and in the denominator part, we will have the term, (4x - 1)^{n+1} (4x - 1) raised to the power of 101 + 1 = 102, as we can see from the obtained pattern. The main part, the numerator is a function of a(n) = 4^n × n!.

So, the function is,

f(x)^n = ± 4^n × n! / (4x - 1)^{n+1}

$f^{101} (x) = - \dfrac{4^{101} \times 101!}{\left(4x-1\right)^{102}}$