Differentiation rules

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

) that return real values, although, more generally, the formulas below apply wherever they are well defined,[1][2] including the case of complex numbers (

This computation shows that the derivative of any constant function is 0.

The derivative of the function at a point is the slope of the line tangent to the curve at the point.

The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0.

In other words, the value of the constant function,

In Leibniz's notation, this formula is written as:

Special cases include:

In Leibniz's notation, this formula is written:

In Leibniz's notation, this formula is written as:

In Leibniz notation, this formula is written as:

, this formula becomes the special case that, if

Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.

In Leibniz's notation, this formula is written:

The elementary power rule generalizes considerably.

The most general power rule is the functional power rule: for any functions

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

Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.

[citation needed] Logarithms can be used to remove exponents, convert products into sums, and convert division into subtraction—each of which may lead to a simplified expression for taking derivatives.

The derivatives in the table above are for when the range of the inverse secant is

and when the range of the inverse cosecant is

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

Suppose that it is required to differentiate with respect to

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

Some rules exist for computing the

consists of all non-negative integer solutions of the Diophantine equation

These rules are given in many books, both on elementary and advanced calculus, in pure and applied mathematics.

Those in this article (in addition to the above references) can be found in: