Английская Википедия:Differentiation rules

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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 <math>c</math>, where <math>c \in \mathbb{R}</math>, if <math>f(x)</math> is the constant function given by <math>f(x) = c</math>, then <math>\frac{df}{dx} = 0</math>.

Proof

Let <math>c \in \mathbb{R}</math> and <math>f(x) = c</math>. By the definition of the derivative,

<math>\begin{align}

f'(x) &= \lim_{h \to 0}\frac{f(x + h) - f(x)}{h} \\ &= \lim_{h \to 0} \frac{(c) - (c)}{h} \\ &= \lim_{h \to 0} \frac{0}{h} \\ &= \lim_{h \to 0} 0 \\ &= 0 \end{align}</math>

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.

Файл:Tangent function animation.gif

At each point, the derivative is the slope of a line that is tangent to the curve at that point. Note: the derivative at point A is positive where green and dash–dot, negative where red and dashed, and zero where black and solid.

Differentiation is linear

For any functions <math>f</math> and <math>g</math> and any real numbers <math>a</math> and <math>b</math>, the derivative of the function <math>h(x) = af(x) + bg(x)</math> with respect to <math>x</math> is: <math> h'(x) = a f'(x) + b g'(x).</math>

In Leibniz's notation this is written as: <math display="block"> \frac{d(af+bg)}{dx} = a\frac{df}{dx} +b\frac{dg}{dx}.</math>

Special cases include:

• The constant factor rule <math display="block">(af)' = af' </math>
• The sum rule <math display="block">(f + g)' = f' + g'</math>
• The difference rule <math display="block">(f - g)' = f' - g'.</math>

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 <math display="block"> h'(x) = (fg)'(x) = f'(x) g(x) + f(x) g'(x).</math> In Leibniz's notation this is written <math display="block">\frac{d(fg)}{dx} = g \frac{df}{dx} + f \frac{dg}{dx}.</math>

The chain rule

The derivative of the function <math>h(x) = f(g(x))</math> is <math display="block"> h'(x) = f'(g(x))\cdot g'(x).</math>

In Leibniz's notation, this is written as: <math display="block">\frac{d}{dx}h(x) = \left.\frac{d}{dz}f(z)\right|_{z=g(x)}\cdot \frac{d}{dx}g(x),</math> often abridged to <math display="block">\frac{dh(x)}{dx} = \frac{df(g(x))}{dg(x)} \cdot \frac{dg(x)}{dx}.</math>

Focusing on the notion of maps, and the differential being a map <math>\text{D}</math>, this is written in a more concise way as: <math display="block"> [\text{D} (f\circ g)]_x = [\text{D} f]_{g(x)} \cdot [\text{D}g]_x\,.</math>

The inverse function rule

If the function Шаблон:Mvar has an inverse function Шаблон:Mvar, meaning that <math>g(f(x)) = x</math> and <math>f(g(y)) = y,</math> then <math display="block">g' = \frac{1}{f'\circ g}.</math>

In Leibniz notation, this is written as <math display="block"> \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}.</math>

Power laws, polynomials, quotients, and reciprocals

The polynomial or elementary power rule

If <math>f(x) = x^r</math>, for any real number <math>r \neq 0,</math> then

<math>f'(x) = rx^{r-1}.</math>

When <math>r = 1,</math> this becomes the special case that if <math>f(x) = x,</math> then <math>f'(x) = 1.</math>

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

The reciprocal rule

Шаблон:Main The derivative of <math>h(x)=\frac{1}{f(x)}</math>for any (nonvanishing) function Шаблон:Mvar is:

<math> h'(x) = -\frac{f'(x)}{(f(x))^2}</math> wherever Шаблон:Mvar is non-zero.

In Leibniz's notation, this is written

<math> \frac{d(1/f)}{dx} = -\frac{1}{f^2}\frac{df}{dx}.</math>

The reciprocal rule can be derived either from the quotient rule, or from the combination of power rule and chain rule.

The quotient rule

Шаблон:Main If Шаблон:Mvar and Шаблон:Mvar are functions, then:

<math>\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}\quad</math> wherever Шаблон:Mvar 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 Шаблон:Mvar and Шаблон:Mvar,

<math>(f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\quad</math>

wherever both sides are well defined.

Special cases

• If <math display="inline">f(x)=x^a\!</math>, then <math display="inline">f'(x)=ax^{a-1}</math> when Шаблон:Mvar is any non-zero real number and Шаблон:Mvar is positive.
• The reciprocal rule may be derived as the special case where <math display="inline">g(x)=-1\!</math>.

Derivatives of exponential and logarithmic functions

<math> \frac{d}{dx}\left(c^{ax}\right) = {ac^{ax} \ln c } ,\qquad c > 0</math>

the equation above is true for all Шаблон:Mvar, but the derivative for <math display="inline">c<0</math> yields a complex number.

<math> \frac{d}{dx}\left(e^{ax}\right) = ae^{ax}</math>

<math> \frac{d}{dx}\left( \log_c x\right) = {1 \over x \ln c} , \qquad c > 1</math>

the equation above is also true for all Шаблон:Mvar, but yields a complex number if <math display="inline">c<0\!</math>.

<math> \frac{d}{dx}\left( \ln x\right) = {1 \over x} ,\qquad x > 0.</math>

<math> \frac{d}{dx}\left( \ln |x|\right) = {1 \over x} ,\qquad x \neq 0.</math>

<math> \frac{d}{dx}\left( W(x)\right) = {1 \over {x+e^{W(x)}}} ,\qquad x > -{1 \over e}.\qquad</math>where <math>W(x)</math> is the Lambert W function

<math> \frac{d}{dx}\left( x^x \right) = x^x(1+\ln x).</math>

<math> \frac{d}{dx}\left( f(x)^{ g(x) } \right ) = g(x)f(x)^{g(x)-1} \frac{df}{dx} + f(x)^{g(x)}\ln{( f(x) )}\frac{dg}{dx}, \qquad \text{if }f(x) > 0, \text{ and if } \frac{df}{dx} \text{ and } \frac{dg}{dx} \text{ exist.}</math>

<math> \frac{d}{dx}\left( f_{1}(x)^{f_{2}(x)^{\left ( ... \right )^{f_{n}(x)}}} \right ) = \left [\sum\limits_{k=1}^{n} \frac{\partial }{\partial x_{k}} \left( f_{1}(x_1)^{f_{2}(x_2)^{\left ( ... \right )^{f_{n}(x_n)}}} \right ) \right ] \biggr\vert_{x_1 = x_2 = ... =x_n = x}, \text{ if } f_{i<n}(x) > 0 \text{ and }</math> <math> \frac{df_{i}}{dx} \text{ exists. }</math>

Logarithmic derivatives

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

<math> (\ln f)'= \frac{f'}{f} \quad</math> wherever Шаблон:Mvar is positive.

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

Derivatives of trigonometric functions

<math> (\sin x)' = \cos x = \frac{e^{ix} + e^{-ix}}{2} </math>	<math> (\arcsin x)' = { 1 \over \sqrt{1 - x^2}} </math>
<math> (\cos x)' = -\sin x = \frac{e^{-ix} - e^{ix}}{2i} </math>	<math> (\arccos x)' = -{1 \over \sqrt{1 - x^2}} </math>
<math> (\tan x)' = \sec^2 x = { 1 \over \cos^2 x} = 1 + \tan^2 x </math>	<math> (\arctan x)' = { 1 \over 1 + x^2} </math>
<math> (\cot x)' = -\csc^2 x = -{ 1 \over \sin^2 x} = -1 - \cot^2 x</math>	<math> (\operatorname{arccot} x)' = {1 \over -1 - x^2} </math>
<math> (\sec x)' = \sec{x}\tan{x} </math>	x|\sqrt{x^2 - 1}} </math>
<math> (\csc x)' = -\csc{x}\cot{x} </math>	<math> (\operatorname{arccsc} x)' = -{1 \over |x|\sqrt{x^2 - 1}} </math>

The derivatives in the table above are for when the range of the inverse secant is <math>[0,\pi]\!</math> and when the range of the inverse cosecant is <math>\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\!</math>.

It is common to additionally define an inverse tangent function with two arguments, <math>\arctan(y,x)\!</math>. Its value lies in the range <math>[-\pi,\pi]\!</math> and reflects the quadrant of the point <math>(x,y)\!</math>. For the first and fourth quadrant (i.e. <math>x > 0\!</math>) one has <math>\arctan(y, x>0) = \arctan(y/x)\!</math>. Its partial derivatives are

<math> \frac{\partial \arctan(y,x)}{\partial y} = \frac{x}{x^2 + y^2}</math>, and <math> \frac{\partial \arctan(y,x)}{\partial x} = \frac{-y}{x^2 + y^2}.</math>

Derivatives of hyperbolic functions

<math>( \sinh x )'= \cosh x = \frac{e^x + e^{-x}}{2}</math>	<math>(\operatorname{arcsinh}x)' = { 1 \over \sqrt{1 + x^2}}</math>
<math>(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}</math>	<math>(\operatorname{arccosh}x)' = {\frac {1}{\sqrt{x^2-1}}}</math>
<math>(\tanh x )'= {\operatorname{sech}^2x} = { 1 \over \cosh^2 x} = 1 - \tanh^2 x</math>	<math>(\operatorname{arctanh}x)' = { 1 \over 1 - x^2}</math>
<math>(\coth x )' = -\operatorname{csch}^2x = -{ 1 \over \sinh^2 x} = 1 - \coth^2 x</math>	<math>(\operatorname{arccoth}x)' =(\operatorname{arctanh}\frac{1}{x})'= \frac {-x^{-2}} {1-\frac {1}{x^2}}=\frac{1}{1-x^2}</math>
<math>(\operatorname{sech} x)' = -\operatorname{sech}{x}\tanh{x}</math>	<math>(\operatorname{arcsech}x)' = -{1 \over x\sqrt{1 - x^2}}</math>
<math>(\operatorname{csch}x)' = -\operatorname{csch}{x}\coth{x}</math>	<math>(\operatorname{arccsch}x)' = -{1 \over |x|\sqrt{1 + x^2}}</math>

See Hyperbolic functions for restrictions on these derivatives.

Derivatives of special functions

Gamma function

<math>\Gamma(x) = \int_0^\infty t^{x-1} e^{-t}\, dt</math>

<math>\begin{align}

\Gamma'(x) & = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt \\ & = \Gamma(x) \left(\sum_{n=1}^\infty \left(\ln\left(1 + \dfrac{1}{n}\right) - \dfrac{1}{x + n}\right) - \dfrac{1}{x}\right) \\ & = \Gamma(x) \psi(x) \end{align}</math> Шаблон:Pb with <math>\psi(x)</math> being the digamma function, expressed by the parenthesized expression to the right of <math>\Gamma(x)</math> in the line above.

Riemann zeta function

<math>\zeta(x) = \sum_{n=1}^\infty \frac{1}{n^x}</math>

<math>\begin{align}

\zeta'(x) & = -\sum_{n=1}^\infty \frac{\ln n}{n^x} =-\frac{\ln 2}{2^x} - \frac{\ln 3}{3^x} - \frac{\ln 4}{4^x} - \cdots \\ & = -\sum_{p \text{ prime}} \frac{p^{-x} \ln p}{(1-p^{-x})^2} \prod_{q \text{ prime}, q \neq p} \frac{1}{1-q^{-x}} \end{align}</math>

Derivatives of integrals

Suppose that it is required to differentiate with respect to x the function

<math>F(x)=\int_{a(x)}^{b(x)}f(x,t)\,dt,</math>

where the functions <math>f(x,t)</math> and <math>\frac{\partial}{\partial x}\,f(x,t)</math> are both continuous in both <math>t</math> and <math>x</math> in some region of the <math>(t,x)</math> plane, including <math>a(x)\leq t\leq b(x),</math> <math>x_0\leq x\leq x_1</math>, and the functions <math>a(x)</math> and <math>b(x)</math> are both continuous and both have continuous derivatives for <math>x_0\leq x\leq x_1</math>. Then for <math>\,x_0\leq x\leq x_1</math>:

<math> F'(x) = f(x,b(x))\,b'(x) - f(x,a(x))\,a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}\, f(x,t)\; dt\,. </math>

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 Шаблон:Mvar-th derivative of functions, where Шаблон:Mvar is a positive integer. These include:

Faà di Bruno's formula

Шаблон:Main If Шаблон:Mvar and Шаблон:Mvar are Шаблон:Mvar-times differentiable, then <math display="block"> \frac{d^n}{d x^n} [f(g(x))]= n! \sum_{\{k_m\}} f^{(r)}(g(x)) \prod_{m=1}^n \frac{1}{k_m!} \left(g^{(m)}(x) \right)^{k_m}</math> where <math display="inline"> r = \sum_{m=1}^{n-1} k_m</math> and the set <math> \{k_m\}</math> consists of all non-negative integer solutions of the Diophantine equation <math display="inline"> \sum_{m=1}^{n} m k_m = n</math>.

General Leibniz rule

Шаблон:Main If Шаблон:Mvar and Шаблон:Mvar are Шаблон:Mvar-times differentiable, then <math display="block"> \frac{d^n}{dx^n}[f(x)g(x)] = \sum_{k=0}^{n} \binom{n}{k} \frac{d^{n-k}}{d x^{n-k}} f(x) \frac{d^k}{d x^k} g(x)</math>

See also

• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link
• Шаблон:Annotated link

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, Шаблон:ISBN.
• The Cambridge Handbook of Physics Formulas, G. Woan, Cambridge University Press, 2010, Шаблон:ISBN.
• Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, Шаблон:ISBN
• NIST Handbook of Mathematical Functions, F. W. J. Olver, D. W. Lozier, R. F. Boisvert, C. W. Clark, Cambridge University Press, 2010, Шаблон:ISBN.

External links

• Derivative calculator with formula simplification