Differentiation of a Function with Respect to Another Function
Let y = f( x) and z = g( x), then the differentiation of y with respect to z is dy / dz = dy / dx / dz / dx = f’( x) / g’( x) Successive Differentiations
If the function y = f( x) be differentiated with respect to x, then the result dy / dx or f’( x), so obtained is a function of x( may be a constant).
Hence, dy / dx can again be differentiated with respect of x.
The differential coefficient of dy / dx with respect to x is written as d / dx( dy / dx) = d 2 y / dx 2 or f’( x). Again, the differential coefficient of d 2 y / dx 2 with respect to x is written as
d / dx( d 2 y / dx 2) = d 3 y / dx 3 or f”'( x)……
Here, dy / dx, d 2 y / dx 2, d 3 y / dx 3,… are respectively known as first, second, third, … order differential coefficients of y with respect to x. These alternatively denoted by f’( x), f”( x), f”’( x), … or y1, y2, y3 …., respectively.
Note dy / dx =( dy / dθ) /( dx / dθ) but d 2 y / dx 2 ≠( d 2 y / dθ 2) /( d 2 x / dθ 2) Leibnitz Theorem If u and v are functions of x such that their nth derivative exist, then