Derivatives, Differentials

Let y = f(x) be a given function. Then, the value of y depends upon the value of x and it changes with a change in the value of x. Let dy be an increment in y, corresponding to an increment dx in x. Then, y = f(x) and y + dy = f(x + dx)

On substitution, we get dy = f(x + dx) - f(x)

Therefore,

The above limit, if it exists finitely, is called the derivative or differential coefficient of y= f(x) with respect to x and it is denoted by

The process of finding the derivative is known as differentiation.

General Rules of Differentiation

In the following, u, v, w are functions of x; a, b, c, n are constants [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) = d²y/dx² = f''(x) = y ''
13.44 Third derivative = (d/dx).(d²y/dx²) = d³/dx³ = f'''(x) = y'''
13.45 n-th derivative = (d/dx).(d^{n - 1}/dx^{n - 1}) = dⁿ/dxⁿ = 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^pu/dx^p = the p-th derivative of u. Then
13.46
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 differentials 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.