# Differentiation rules

This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.

## Elementary rules of differentiation

Unless otherwise stated, all functions are functions of real numbers (R) that return real values; although more generally, the formulae below apply wherever they are well defined[1][2]—including complex numbers (C).[3]

### Differentiation is linear

For any functions and and any real numbers and the derivative of the function with respect to is

In Leibniz's notation this is written as:

Special cases include:

• The subtraction rule

### The product rule

Main article: Product rule

For the functions f and g, the derivative of the function h(x) = f(x) g(x) with respect to x is

In Leibniz's notation this is written

### The chain rule

Main article: Chain rule

The derivative of the function with respect to is

In Leibniz's notation this is correctly written as:

often abridged to Focusing on the notion of maps, and the differential being a map , this is written in a more concise way as:

### The inverse function rule

If the function f has an inverse function g, meaning that g(f(x)) = x and f(g(y)) = y, then

In Leibniz notation, this is written as

## Power laws, polynomials, quotients, and reciprocals

### The polynomial or elementary power rule

Main article: Power rule

If , for any real number then

Special cases include:

• If f(x) = x, then f′(x) = 1. This special case may be generalized to:
The derivative of an affine function is constant: if f(x) = ax + b, then f′(x) = a.

Combining this rule with the linearity of the derivative and the addition rule permits the computation of the derivative of any polynomial.

### The reciprocal rule

Main article: Reciprocal rule

The derivative of h(x) = 1/f(x) for any (nonvanishing) function f is:

In Leibniz's notation, this is written

The reciprocal rule can be derived from the quotient rule.

### The quotient rule

Main article: Quotient rule

If f and g are functions, then:

wherever g is nonzero.

This can be derived from product rule.

### Generalized power rule

Main article: Power rule

The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions f and g,

wherever both sides are well defined.

Special cases:

• If f(x) = xa, f′(x) = axa − 1 when a is any non-zero real number and x is positive.
• The reciprocal rule may be derived as the special case where g(x) = −1.

## Derivatives of exponential and logarithmic functions

note that the equation above is true for all c, but the derivative for c < 0 yields a complex number.

the equation above is also true for all c but yields a complex number if c<0.

### Logarithmic derivatives

The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):

wherever f is positive.

## Derivatives of trigonometric functions

It is common to additionally define an inverse tangent function with two arguments, . Its value lies in the range and reflects the quadrant of the point . For the first and fourth quadrant (i.e. ) one has . Its partial derivatives are

 , and

## Derivatives of special functions

 Gamma function with being the digamma function, expressed by the parenthesized expression to the right of in the line above.

## Derivatives of integrals

Suppose that it is required to differentiate with respect to x the function

where the functions and are both continuous in both and in some region of the plane, including , and the functions and are both continuous and both have continuous derivatives for . Then for :

This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.

## Derivatives to nth order

Some rules exist for computing the nth derivative of functions, where n is a positive integer. These include:

### Faà di Bruno's formula

If f and g are n times differentiable, then

where and the set consists of all non-negative integer solutions of the Diophantine equation .

### General Leibniz rule

Main article: General Leibniz rule

If f and g are n times differentiable, then

## References

1. Calculus (5th edition), F. Ayres, E. Mendelson, Schuam's Outline Series, 2009, ISBN 978-0-07-150861-2.
2. Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schuam's Outline Series, 2010, ISBN 978-0-07-162366-7.
3. Complex Variables, M.R. Speigel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, ISBN 978-0-07-161569-3