The function is described as $$f_p(x) = \dfrac{9\sqrt{x^2+p}}{x^2+2}$$ We need to differentiate w.r.t. $x$
One solution is: $$f_p^2(x) (x^2 + 2)^2 = 81(x^2 + p) \quad\Longrightarrow$$
$$2 f_p(x) f_p'(x)(x^2 + 2)^2 + 4x f_p^2(x^2 +2) = 162x \quad\Longrightarrow$$\begin{align}f_p'(x) &= \frac{162x - 4x f_p^2(x)(x^2+2)}{2f_p(x)(x^2+2)^2}\\ \\ &= \frac{162x - 324x\frac{x^2 + p}{x^2 +2}}{18(x^2 +2)\sqrt{x^2+p}}\\ \\ &= \frac{9 x (x^2+2) - 18x(x^2+p)}{(x^2 + 2)^2 \sqrt{x^2 + p}}\\ \\ &= \frac{9x(2-2p-x^2)}{(x^2 + 2)^2\sqrt{x^2 + p}}\end{align}We will attempt to solve this using features in SymEngine
In [1]:
require 'symengine'
Out[1]:
Declare the symbols
In [3]:
x = SymEngine::Symbol.new('x')
p = SymEngine::Symbol.new('p')
half = 1.quo(2)
Out[3]:
Create the expression
In [5]:
fp = 9*((x**2 + p)**half)/((x**2)+2)
fp.to_s
Out[5]:
Differentiate wrt $x$
In [8]:
answer = fp.diff(x)
answer.to_s
Out[8]:
Which is indeed correct! If you simplify the second last answer that we got from our solution. You will find the exact same answer. Using SymEngine for solving problems is as simple as that.