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DIFFERENTIATION RULES AND FORMULAS

Rules and formulas for differentiation. The rules and formulas for finding the derivatives of functions of a complex variable are identical to those for real variables. The various formulas are derived in exactly the same way as for real variables. One starts from the basic definition

and uses the various theorems on limits to arrive at the formulas.

In the following rules and formulas u and v are differentiable functions of z while a and c are constants.

The derivative of a constant is zero.

The derivative of a variable with respect to itself is one.

The derivative of the sum of two functions is equal to the sum of their separate derivatives.

The derivative of the product of two functions is the first function times the derivative of the second plus the second function times the derivative of the first.

The derivative of the product of a constant and a function is the constant times the derivative of the function.

The derivative of the quotient of a function by a constant is the derivative of the function divided by the constant.

The derivative of w with respect to z is the reciprocal of the derivative of z with respect to w.

The derivative of the quotient of two functions is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

9. Let w be a function of u and u be a function of z i.e. w = f(u) and u = g(z). Then

where n is any real number (positive, negative, fractional, etc).

The derivative of the n-th power of z is n times the (n-1)th power of z.

where n is any real number (positive, negative, fractional, etc).

The derivative of the n-th power of a function is n times the (n-1)th power of the function times the derivative of the function.

Trigonometric functions

Inverse trigonometric functions

Exponential and logarithmic functions

where a is a constant.

Hyperbolic functions

Inverse hyperbolic functions