Heat said:
Homework Statement
Find d^2y/dx^2.
y = x cos x
First find the first derivative. You get two terms (product rule!).
What is the difference of dx/dy, d^2y/dx^2, dy^2/d^2x, etc...
dx/dy is the derivative of
x with respect to
y. Writing this down sort of implies that
x is a function of
y, like x(y) = 3 \sqrt{1 + y^2} / y, though it's a bit against conventions to call the function
x and the variable
y.
d^2 y / dx^2 is the second derivative of y with respect to x, of which y is a function. It can be viewed as shorthand for
\frac{d^2 y}{dx^2} = \frac{d\left( \frac{ dy }{ dx } \right) }{ dx }
Of course this is possible, since dy/dx is again a function of
x, for example:
y(x) = x^2, \frac{dy}{dx} = 2 x, \frac{d^2 y}{dx} = \frac{d (2x) }{dx} = 2.
Oh, and dy^2/dx^2 doesn't really mean anything, it's usually an error if someone writes that down. (Though it is possible to make sense of it, by viewing x^2 as a variable, e.g. y = 2 x^2 s.t. y^2 = 4 (x^2)^2 = 4 x^4. Then \frac{ d y^2 }{ d x^2 } = 8 x^2 as you can see by temporarily writing z = x^2 everywhere, deriving, and writing it back in x^2).
