From Wikipedia, the free encyclopedia - View original article

Library resources aboutDifferentiation rules |

Calculus |
---|

Specialized |

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

- 1 Elementary rules of differentiation
- 2 Power laws, polynomials, quotients, and reciprocals
- 3 Derivatives of exponential and logarithmic functions
- 4 Derivatives of trigonometric functions
- 5 Derivatives of hyperbolic functions
- 6 Derivatives of special functions
- 7 Derivatives of integrals
- 8 Derivatives to
*n*th order - 9 See also
- 10 References
- 11 Sources and further reading
- 12 External links

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]}

Main article: Linearity of differentiation

For any functions *f* and *g* and any real numbers *a* and *b* the derivative of the function *h*(*x*) = *af*(*x*) + *bg*(*x*) with respect to *x* is

In Leibniz's notation this is written as:

Special cases include:

*The sum rule*

*The subtraction 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

Main article: Chain rule

The derivative of the function of a function *h*(*x*) = *f*(*g*(*x*)) with respect to *x* is

In Leibniz's notation this is written as:

However, by relaxing the interpretation of *h* as a function, this is often simply written

Main article: inverse functions and differentiation

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

Main article: Power rule

If , for any integer *n* then

Special cases include:

*Constant rule*: if*f*is the constant function*f*(*x*) =*c*, for any number*c*, then for all*x*,*f′*(*x*) = 0.- 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.

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 chain rule and the power rule.

Main article: Quotient rule

If *f* and *g* are functions, then:

- wherever
*g*is nonzero.

This can be derived from reciprocal rule and the product rule. Conversely (using the constant rule) the reciprocal rule may be derived from the special case *f*(*x*) = 1.

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*) =*x*^{a},*f′*(*x*) =*ax*^{a − 1}when*a*is any real number and*x*is positive. - The reciprocal rule may be derived as the special case where
*g*(*x*) = −1.

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.

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.

Main article: Differentiation 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 |

Main article: Differentiation under the integral sign

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.

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

Main article: 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 .

Main article: General Leibniz rule

If *f* and *g* are *n* times differentiable, then

- Derivative
- Differential calculus
- Vector calculus identities
- Differentiable function
- Differential of a function
- Limit of a function
- Function (mathematics)
- List of mathematical functions
- Trigonometric functions
- Inverse trigonometric functions
- Hyperbolic functions
- Inverse hyperbolic functions
- Matrix calculus
- Differentiation under the integral sign

**^***Calculus (5th edition)*, F. Ayres, E. Mendelson, Schuam's Outline Series, 2009, ISBN 978-0-07-150861-2.**^***Advanced Calculus (3rd edition)*, R. Wrede, M.R. Spiegel, Schuam's Outline Series, 2010, ISBN 978-0-07-162366-7.**^***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

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, Schuam's Outline Series, 2009, ISBN 978-0-07-154855-7.*The Cambridge Handbook of Physics Formulas*, G. Woan, Cambridge University Press, 2010, ISBN 978-0-521-57507-2.*Mathematical methods for physics and engineering*, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3*NIST Handbook of Mathematical Functions*, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010, ISBN 978-0-521-19225-5.

Library resources aboutDifferentiation rules |