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 the case of
complex numbers (C).[3]
Constant term rule
For any value of , where , if is the constant function given by , then .[4]
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 its angle is zero.
In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
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.
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 For the first and fourth quadrant (i.e. ) one has Its partial derivatives are
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 :
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,
ISBN978-0-07-154855-7.
The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010,
ISBN978-0-521-57507-2.
Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010,
ISBN978-0-521-86153-3
NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010,
ISBN978-0-521-19225-5.
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 the case of
complex numbers (C).[3]
Constant term rule
For any value of , where , if is the constant function given by , then .[4]
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 its angle is zero.
In other words, the value of the constant function, y, will not change as the value of x increases or decreases.
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.
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 For the first and fourth quadrant (i.e. ) one has Its partial derivatives are
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 :
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,
ISBN978-0-07-154855-7.
The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010,
ISBN978-0-521-57507-2.
Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010,
ISBN978-0-521-86153-3
NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010,
ISBN978-0-521-19225-5.