Differentiation rules

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 — including the case of complex numbers (C).
Constant term rule
For any value of , where , if is the constant function given by , then .
Proof
Let and . By the definition of the derivative,
This shows that the derivative of any constant function is 0.
Intuitive (geometric) explanation
The derivative of the function at a point is the slope of the line tangent to the curve at the point. Slope of the constant function is zero, because the tangent line to the constant function is horizontal and it's angle is zero.
In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
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 constant factor rule
The sum rule
The difference rule
The 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
The derivative of the function is
In Leibniz's notation, this is 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 has an inverse function , meaning that and then
In Leibniz notation, this is written as
Power laws, polynomials, quotients, and reciprocals
The polynomial or elementary power rule
If , for any real number then
When this becomes the special case that if then
Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.
The reciprocal rule
The derivative of for any (nonvanishing) function is:
wherever is non-zero.
In Leibniz's notation, this is written
The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule.
The quotient rule
If and are functions, then:
wherever is nonzero.
This can be derived from the product rule and the reciprocal rule.
Generalized power rule
The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions and ,
wherever both sides are well defined.
Special cases
If , then when is any non-zero real number and is positive.
The reciprocal rule may be derived as the special case where .
Derivatives of exponential and logarithmic functions
the equation above is true for all , but the derivative for yields a complex number.
the equation above is also true for all , but yields a complex number if .
where is the Lambert W function
Logarithmic derivatives
The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):
wherever is positive.
Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
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.
Derivatives of trigonometric functions
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 . For the first and fourth quadrant (i.e. ) one has . Its partial derivatives are
Derivatives of hyperbolic functions
See Hyperbolic functions for restrictions on these derivatives.
Derivatives of special functions
Gamma function
with being the digamma function, expressed by the parenthesized expression to the right of in the line above.
Riemann zeta function
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 -th derivative of functions, where is a positive integer. These include:
Faà di Bruno's formula
If and are -times differentiable, then
where and the set consists of all non-negative integer solutions of the Diophantine equation .
General Leibniz rule
If and are -times differentiable, then
See also
References
Sources and further reading
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:
Mathematical Handbook of Formulas and Tables (3rd edition), S. Lipschutz, M.R. Spiegel, J. Liu, Schaum's Outline Series, 2009, .
The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, .
Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010,
NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010, .
External links
Derivative calculator with formula simplification
Articles containing proofs
Derivatives
Derivatives
Mathematical identities
Theorems in analysis
Theorems in calculus