# Thread: Chain rule question #3

1. ## Chain rule question #3

if you have

y=SQUAREROOT (x^2 + 3)

let z = (x^2 + 3)

...1 dy/dz= 1/2 z ^-1/2 and dz/dx= 2 x <<< can someone explain where the 2 is derived from?

...2 then dy/dz x dz/dx = 1/2 z ^-1/2 (2 x)

...3 this = x/squareroot z = x/SQUAREROOT (x^2 + 3)

how do you get from line 2 to line 3 as x/SQUAREROOT (x^2 + 3)

2. Line 1: 2x comes from taking the derivative of x^2+3 which the chain rule requires.

From line 2 you have $\dfrac{dy}{dx} = \dfrac{1}{2} (x^2+3)^{-1/2}$

From the laws of exponents and surds we know that $a^{-1/n} = \dfrac{1}{\sqrt[n]{a}}$

Thus $\dfrac{1}{2} (x^2+3)^{-1/2} = \dfrac{1}{2} \cdot \dfrac{1}{\sqrt{x^2+3}}$

When multiplying by 2x the 1/2 will cancel leaving you with $\dfrac{x}{\sqrt{x^2+3}}$

Which is line 3