I need help with differentiation.
I don't quite understand what is derivative of y with respect to some function. Yes x is also a function but let me show the example.
if y = x^2 + 2, what is derivative of y with respect to (1-x) ?
I need help with differentiation.
I don't quite understand what is derivative of y with respect to some function. Yes x is also a function but let me show the example.
if y = x^2 + 2, what is derivative of y with respect to (1-x) ?
I've never heard of such a thing. Are you perhaps supposed to do some sort of substitution, like "u = 1 - x"? So then x = 1 - u, and x^2 + 2 = u^2 - 2u + 3? And then you differentiate with respect to u, and then maybe back-substitute?
I'm just guessing....
You can differentiate any (differentiable) function with respect to any other (differentiable) function. Let f(x) and g(x) be differentiable functions of x. By the chain rule, $\displaystyle \frac{df}{dg}= \frac{df}{dx}\frac{dx}{dg}$$\displaystyle = \frac{df}{dx}\frac{1}{\frac{dg}{dx}}= \frac{\frac{df}{dx}}{\frac{dg}{dx}}$.
Here $\displaystyle f(x)= x^2+ 2$ and $\displaystyle g(x)= 1- x$. $\displaystyle \frac{df}{dx}= 2x$ and $\displaystyle \frac{dg}{dx}= -1$ so $\displaystyle \frac{d(x^2+2)}{d(1-x)}= \frac{df}{dg}= \frac{2x}{-1}= -2x$.
The derivative of f with respect to g essentially tells how fast f is increasing or decreasing compared to how fast g is.
Exactly! Thanks.
I substitute and back subsitute.
$\displaystyle \frac{d}{d(x-1)} y$
$\displaystyle y = x^2+1$
and then substituting:
$\displaystyle u = x-1, x=1+u$
and with chain rule I get:
$\displaystyle \frac{d}{du} (u+1)^2+1 = D((u+1)^2) \times D(u+1) = 2(u+1)\times1$
back substitute x:
$\displaystyle 2(1-x)+1) = 2-4x$
Thanks but hmm...you get -2x ?
EQUATION UPDATED. My mistake earlier.
I substitute and back subsitute.
$\displaystyle \frac{d}{d(1-x)} y$
$\displaystyle y = x^2+1$
and then substituting:
$\displaystyle u = 1-x, x=1-u$
and with chain rule I get:
$\displaystyle \frac{d}{du} (1-u)^2+1 = D((1-u)^2) \times D(1-u) = 2(1-u)\times1$
back substitute x:
$\displaystyle 2(1-1-x) = -2x$
HallsofIvy, your example is much more faster, I have to study that.
Thank you guys!!