Definition of a Derivative
If y = f(x), the derivative of y or f(x) with respect to x is defined as
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
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
where are the binomial coefficients.
As special cases we have
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
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
Similarly the partial derivative of f(x, y) with respect to y, keeping x constant, is defined to be
Partial derivatives of higher order can be defined as follows.
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
where dx = Δx and dy = Δy.
Extension to functions of more than two variables are exactly analogous.