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