TopA function is that mathematical term which states relationship between constants and one or more variables. For example, consider a function f(x) = 7x4 + 100, which expresses a relationship between the variable ‘x’ that is raised to some power 4 and constants 7 and 100. We are aware of the word differentiation and its use. It can also be called as derivative and is represented as dy / dx or f'(x). While differentiating exponential Functions we need to follow a formula that is given as follows: d (ax) /d x = x ax-1. The same rule is applied for finding the further Derivatives also.
To understand it better let us consider an example of a function that is given as::
F (v) = 10 v7 – 41 v6 + 4 v4 + 15 v2 + 10 v
To differentiate this function we will follow the above mentioned formula and the general rules of differentiation. So, we can write the derivative with respect to the variable v as:
F’ (v) = d (10 v7 – 41 v6 + 4 v4 + 15 v2 + 10 v) / dv, differentiating each term in the function with respect to v:
F’ (v) = d (10 v7) / dv – d (41 v6) / dv + d (4 v4) / dv+ d (15 v2) / dv+ d (10 v) / dv
As the differentiation of constant values results to zero we are left with:
F’ (v) = 10 d (v7) / dv – 41 d (v6) / dv + 4 d (v4) / dv+ 15 d (v2) / dv+ 10 d (v) / dv
Using the formula we get:
F’ (v) = (7 * 10 (v7-1)) – (6 * 41 (v6-1)) + (4 * 4 (v4-1)) + (15 * 2 (v2-1)) + (10 *1 (v1-1))
Or F’ (v) = 70 v6 – 246 v5 + 16 v3 + 30 v1 + 10 (First derivative)
In the similar way the other higher order derivatives can be finding easily.