Question

Can someone please explain the chain rule to me?

Answer 1

The chain rule lets you take derivatives of functions more complicated than the basic ones you've been shown.

For instance you can probably take the derivative of
\[y=x^6\to y^{\prime}=6x^5\]
But what do you do when you have
\[y=(5x^3+3x^2+23)^6\]

Answer 2

The chain rule allows you to substitute a function of x for the interior of that function above
\[y=(5x^3+3x^2+23)^6\to (u(x))^6\]

Because we know that \[\frac{d}{du} u^6=6u^5\]

Answer 3

it gets confusing so i'm going to give you the way i understand it best.
let's say we have \[\sqrt{x^2+1}\]
we can re write this as \[(x+1)^\frac{ 1 }{ 2 }\]
first we take the derivitive of the outer function which is \[\sqrt{?}\]
so that is \[\frac{ 1 }{2 }(the stuff)^{\frac{ -1 }{ 2 }}\]
or \[\frac{ 1 }{2 }(x+1)^{\frac{ -1 }{ 2 }}\]
then we multiply that by the derivitave of the stuff
\[\frac{ 1 }{2 }(x+1)^{\frac{ -1 }{ 2 }} * 1\]