# Derivatives, Differentials

#### Definition of a Derivative

If y = f(x), the derivative of y or f(x) with respect to x is defined as

13.1

where h = Δx. The derivative is also denoted by y', df/dx of f'(x). The process of taking a derivative is called *differentiation*.

**General Rules of Differentiation**

In the following, u, v, w are functions of x; a, b, c, n are constanst [restricted if indicated]; e = 2.71828... is the natural base of logarithms; ln u is the natural logarithm of u [i.e. the logarithm to the base e] where it is assumed that u > 0 and all angles are in radians.

#### Derivatives of Trigonometric and Inverse Trigonometric Functions

#### Derivatives of Exponential and Logarithmic Functions

**Derivatives of Hyperbolic and Inverse Hyperbolic Functions**

#### Higher Derivatives

The second, third and higher derivatives are defined as follows.

13.43 Second derivative = (d/dx).(dy/dx) = d^{2}y/dx^{2} = f''(x) = y ''

13.44 Third derivative = (d/dx).(d^{2}y/dx^{2}) = d^{3}/dx^{3} = f'''(x) = y'''

13.45 n-th derivative = (d/dx).(d^{n - 1}/dx^{n - 1}) = d^{n}/dx^{n} = f^{(n)}(x) = y^{(n)}

**Leibnitz's Rule for Higher Derivatives of Products**

Let D^{p} stand for the operator d^{p}/dx^{p} so that D^{P} u = d^{p}u/dx^{p} = the p-th derivative of u. Then

13.46

where are the binomial coefficients.

As special cases we have

13.47

13.48

#### Differentials

Let y = f(x) and Δy = f(x + Δx) - f(x). Then

13.49 Δy/Δx = [f(x + Δx) - f(x)]/Δx = f'(x) + ε = dy/dx + ε

where ε → 0 as Δx → 0. Thus

13.50 Δy = f'(x)Δx + εΔx

If we call Δx = dx the differential of x, then we define the differential of y to be

13.51 dy = f'(x)dx

#### Rules for Differentials

The rules for differetials are exactly analogous to those for derivatives. As examples we observe that

#### Partial Derivatives

Let f(x, y) be a function of the two variables x and y. Then we define the partial derivative of f(x, y) with respect to x, keeping y constant, to be

13.58

Similarly the partial derivative of f(x, y) with respect to y, keeping x constant, is defined to be

13.59

Partial derivatives of higher order can be defined as follows.

13.60

13.61

The results in 13.61 will be equal if the function and its partial derivatives are continuous, i.e. in such case the order of differentiation makes no difference.

The differential of f(x, y) is defined as

13.62

where dx = Δx and dy = Δy.

Extension to functions of more than two variables are exactly analogous.