TopDerivatives of exponential Functions are nothing but differentiating those mathematical terms that involve powers. Functions in the form f (a) = h a4 + c a2 + a are called exponential functions. Where, h and c are the constants that are in a relationship with the variable a, with respect to which the functions has to be differentiated. Taking derivative of an Exponential Function follows a general formula: d (xn) /d x = n xn-1. The higher order Derivatives can also be determine using the same formula and also applying other general rules of addition, subtraction, multiplication and division of differentiation.
Let’s consider an example to show you how it works: Say we have a function, F (x) = x4 – x3 + x2 + x. Finding the derivative of this function with respect to the variable x means finding the derivative of individual terms in the function. So, by doing it we get:
F’ (x) = d (x4 – x3 + x2 + x) / dx,
F’ (x) = d (x4) / dx – d (x3) / dx + d (x2) / dx+ d (x) / dx,
F’ (x) =4 (x4-1) – 3 (x3-1)) + 2 (x2-1)) + x1-1,
Or F’ (x) = 4 x3 – 3 x2 + 2 x + 1 (First derivative).
Finding the second derivative of the function that we get as the first derivative:
F’’ (x) = d (4 x3 – 3 x2 + 2 x +1) / dx,
F’’ (x) = 4 d (x3) / dx – 3 d (x2) / dx + 2 d (x) / dx+ d (1) / dx,
F’’ (x) =12 (x3-1) – 6 (x2-1)) + 2 (x1-1)) + 0,
Or F’ (x) = 12 x2 – 6 x + 2 (Second derivative).
It is noticeable that as we go on differentiating the function, it goes on reducing. In the similar way, other higher order derivatives can be determined.