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) = 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).(dn - 1/dxn - 1) = dn/dxn = f(n)(x) = y(n)

Leibnitz's Rule for Higher Derivatives of Products

Let Dp stand for the operator dp/dxp so that DP u = dpu/dxp = 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.

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