differentiate wid respect to (x^2+1)^1/2?

Question

f(x)=ln(x^2+34)+x^3+ 2x differentiate wid respect to
(x^2+1)^1/2

Answer 1

f(x)=ln(x^2+34)+x^3+ 2x differentiate wid respect to
let
z=(x^2+1)^1/2

df/dx = [2x/(x^2+34)] +3 x^2 +2

dz/dx = x (x^2+1)^-1/2

df/dz = df/dx * dx/dz
= {[2x/(x^2+34)] +3 x^2 +2 } *{(x^2+1)^1/2 /x}

= {[2/(x^2+34)] +3 x^2 +2 } *{(x^2+1)^1/2}

={(x^2+1)^1/2} / (x^2+34)} {3x^4 +104 x^2 +70}

Answer 2

the entire point of this exercise is using implicit differentiation.

first let u = √(x² + 1) and differentiate with respect to u.

y = ln(x² + 34) + x³ + 2x

dy/du = (2xdx/du)/( x² + 34) + 3x² dx/du + 2dx/du

factor out dx/du:

dy/du = dx/du[ 2x/( x² + 34) + 3x² + 2)]


now, since u = √(x² + 1) --> x = √(u² - 1) -->dx/du = u/√(u² - 1)

so

dy/du = u/√(u² - 1)[ 2x/( x² + 34) + 3x² + 2)]

substitute for u = √(x² + 1)

dy/d{√(x² + 1)} = [√(x² + 1)/x][ 2x/( x² + 34) + 3x² + 2)]

Answer 3

x^1+x^1+0.5