List of Notes - Category: Higher-Order Derivatives
If `f(x)` is a differentiable function, then its derivative `f'(x)` is also a function, so may have a derivative (finite or not). This function is called second derivative of `f(x)` because it is derivative of derivative and denoted by `f''` . So, `f''=(f')'`.
In general, to find n-th derivative of function `y=f(x)` we need to find all derivatives of previous orders. But sometimes it is possible to obtain expression for n-th derivative that depends on `n` and doesn't contain previous derivatives.
Differential of the second order of function `y=f(x)` is differential of first differential of the function: `d^2y=d(dy)`.
Differential of the third order of function `y=f(x)` is differential of second differential of the function: `d^3y=d(d^2y)`.
Sometimes we need to write derivatives with respect to `x` through differentials of another variable `t`. In this case expression for derivatives will be more complex.
So, let's calculate differentials with respect to `t`, in other words `x` is not an independent variable.