# Differential Calculus-i

**Differential Calculus-1:
**

The mathematical study of change like motion, growth or decay is calculus. The Rate of change of given function is derivative or differential. The concept of derivative is essential in day to day life. Also applicable in Engineering, Science, Economics, Medicine etc.

**Successive Differentiation: **

Let y = f (x) --(1) be a real valued function.The first order derivative of y denoted by or y’ or y_{1} or∆1. The Second order derivative of y denoted by or y’’ or y_{2} or ∆2. Similarly differentiating the function (1) n-times, successively, the n^{th}order derivative of y exists denoted by or y_{n} or y_{n} or ∆n. The process of finding 2^{nd }and higher order derivatives is known as Successive Differentiation.

**n ^{th}derivative of some standard functions:**

**1.** y = e^{ax}

**Solutions:** y_{1} = a e^{ax}

y_{2} = a^{2} e^{ax}

Differentiate Successively we get;

y_{n} = a_{n }e^{ax}

**2**.

**Solutions:**

**3.
**

Similarly,

**Case -1** If m = n in (*)

**Case -2** If m > n in (*)