Wikipedia / Differentiation rules
Wikipedia / Differentiation rules
This article 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) that return real values, although, more generally, the formulas below apply wherever they are well defined, including the case of complex numbers).
Constant term rule
.
Proof
This computation 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. The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0.
increases or decreases.
Linearity of differentiation
.
Special cases include:
The constant factor rule:
The sum rule:
The difference rule:
Product rule
Chain rule
Inverse function rule
has an inverse function
Power laws, polynomials, quotients, and reciprocals
Polynomial or elementary power rule
.
Combining the power rule with the sum and constant multiple rules permits the computation of the derivative of any polynomial.
Reciprocal rule
is nonzero.
The reciprocal rule can be derived either from the quotient rule or from the combination of power rule and chain rule.
Quotient rule
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 wherever both sides are well defined.
Special cases:
is positive.
.
Derivatives of exponential and logarithmic functions
yields a complex number.
.
is the Lambert W function.
Logarithmic derivatives
The logarithmic derivative is another way of stating the rule for differentiating the logarithm 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
.
It is common to additionally define an inverse tangent function with two arguments
Derivatives of hyperbolic functions
Derivatives of special functions
Gamma function
being the digamma function in the line above.
Riemann zeta function
Derivatives of integrals
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
is a positive integer, including:
Faà di Bruno's formula
consists of all non-negative integer solutions of the Diophantine equation.
General Leibniz rule
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