Differentiation rules

Topics in Calculus

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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 will be functions from R to R, although more generally, the formulae below make sense wherever they are well defined.

Differentiation is linear

Main article: Linearity of differentiation

For any functions f and g and any real numbers a and b.

$(af + bg)' = af' + bg'.\,$

In other words, the derivative of the function h(x) = a f(x) + b g(x) with respect to x is

$h'(x) = a f'(x) + b g'(x).\,$

In Leibniz's notation this is written

$\frac{d(af+bg)}{dx} = a\frac{df}{dx} +b\frac{dg}{dx}.$

Special cases include:

• The constant multiple rule

$(af)' = a\,f' \,$

• The sum rule

$(f + g)' = f' + g'\,$

• The subtraction rule

$(f - g)' = f' - g'.\,$

The product or Leibniz rule

Main article: Product rule

For any of the functions f and g,

$(fg)' = f' g + f g'.\,$

In other words, the derivative of the function h(x) = f(x) g(x) with respect to x is

$h'(x) = f'(x) g(x) + f(x) g'(x).\,$

In Leibniz's notation this is written

$\frac{d(fg)}{dx} = \frac{df}{dx} g + f \frac{dg}{dx}.$

The chain rule

Main article: Chain rule

This is a rule for computing the derivative of a function of a function, i.e., of the composite $f\circ g$ of two functions f and g:

$(f \circ g)' = (f' \circ g)g'.\,$

In other words, the derivative of the function h(x) = f(g(x)) with respect to x is

$h'(x) = f'(g(x)) g'(x).\,$

In Leibniz's notation this is written (suggestively) as:

$\frac{df}{dx} = \frac{df}{dg} \frac{dg}{dx}.\,$

The polynomial or elementary power rule

Main article: Calculus with polynomials

If f(x) = xⁿ, for some natural number n (including zero) then

$f'(x) = nx^{n-1}.\,$

Special cases include:

• Constant rule: if f is the constant function f(x) = c, for any number c, then for all x

$f'(x) = 0.\,$

• The derivative of a linear function is constant: if f(x) = ax (or more generally, in view of the constant rule, if f(x)=ax+b ), then

$f'(x) = a.\,$

Combining this rule with the linearity of the derivative permits the computation of the derivative of any polynomial.

The reciprocal rule

Main article: Reciprocal rule

not

For any (nonvanishing) function f, the derivative of the function 1/f (equal at x to 1/f(x)) is

$-\frac{f'}{f^2}.\,$

In other words, the derivative of h(x) = 1/f(x) is

$h'(x) = -\frac{f'(x)}{[f(x)]^2}.\$

In Leibniz's notation, this is written

$\frac{d(1/f)}{dx} = -\frac{1}{f^2}\frac{df}{dx}.\,$

The inverse function rule

Main article: inverse functions and differentiation

This should not be confused with the reciprocal rule: the reciprocal 1/x of a nonzero real number x is its inverse with respect to multiplication, whereas the inverse of a function is its inverse with respect to function composition.

If the function f has an inverse g = f⁻¹ (so that g(f(x)) = x and f(g(y)) = y) then

$g' = \frac{1}{f'\circ f^{-1}}.\,$

In Leibniz notation, this is written (suggestively) as

$\frac{dx}{dy} = \frac{1}{dy/dx}.$

Further rules of differentiation

The quotient rule

Main article: Quotient rule

If f and g are functions, then:

$\left(\frac{f}{g}\right)' = \frac{f'g - g'f}{g^2}\quad$ wherever g is nonzero.

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

Generalized power rule

Main article: Power rules

The elementary power rule generalizes considerably. First, if x is positive, it holds when n is any real number. The reciprocal rule is then the special case n = -1 (although care must then be taken to avoid confusion with the inverse rule).

The most general power rule is the functional power rule: for any functions f and g,

$(f^g)' = \left(e^{g\ln f}\right)' = f^g\left(f'{g \over f} + g'\ln f\right),\quad$

wherever both sides are well defined.

Logarithmic derivatives

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

$(\ln f)'= \frac{f'}{f} \quad$ wherever f is positive.

Derivatives of exponential and logarithmic functions

$\left(c^{ax}\right)' = {c^{ax} \ln c \cdot a } ,\qquad c > 0$

note that the equation above is true for all c, but the derivative for c < 0 yields a complex number.

$\left(e^x\right)' = e^x$

$\left( \log_c x\right)' = {1 \over x \ln c} , \qquad c > 0, c \ne 1$

the equation above is also true for all c but yields a complex number.

$\left( \ln x\right)' = {x' \over x} ,\qquad x \ne 0$

$\left( \ln |x|\right)' = {x' \over x}$

$\left( x^x \right)' = x^x(1+\ln x)$

The derivative of the natural logarithm with a generalised functional argument f(x) is

$\frac{d}{dx}[\ln(f(x))] = \frac{f'(x)}{f(x)}$

By applying the change-of-base rule, the derivative for other bases is

$\frac{d}{dx} \log_b(x) = \frac{d}{dx} \frac {\ln(x)}{\ln(b)} = \frac{1}{x \ln(b)} = \frac{\log_b(e)}{x}.$

Derivatives of trigonometric functions

For more details on this topic, see Differentiation of trigonometric functions.

$(\sin x)' = \cos x \,$	$(\sin^{-1} x)' = { 1 \over \sqrt{1 - x^2}} \,$
$(\cos x)' = -\sin x \,$	$(\cos^{-1} x)' = -{1 \over \sqrt{1 - x^2}} \,$
$(\tan x)' = \sec^2 x = { 1 \over \cos^2 x} = 1 + \tan^2 x \,$	$(\tan^{-1} x)' = { 1 \over 1 + x^2} \,$
$(\sec x)' = \sec x \tan x \,$	$(\sec^{-1} x)' = { 1 \over |x|\sqrt{x^2 - 1}} \,$
$(\csc x)' = -\csc x \cot x \,$	$(\csc^{-1} x)' = -{1 \over |x|\sqrt{x^2 - 1}} \,$
$(\cot x)' = -\csc^2 x = { -1 \over \sin^2 x} = -(1 + \cot^2 x)\,$	$(\cot^{-1} x)' = -{1 \over 1 + x^2} \,$

Derivatives of hyperbolic functions

$( \sinh x )'= \cosh x = \frac{e^x + e^{-x}}{2}$	$(\operatorname{arsinh}\,x)' = { 1 \over \sqrt{x^2 + 1}}$
$(\cosh x )'= \sinh x = \frac{e^x - e^{-x}}{2}$	$(\operatorname{arcosh}\,x)' = {\frac {1}{\sqrt{x^2-1}}}$
$(\tanh x )'= {\operatorname{sech}^2\,x}$	$(\operatorname{artanh}\,x)' = { 1 \over 1 - x^2}$
$(\operatorname{sech}\,x)' = - \tanh x\,\operatorname{sech}\,x$	$(\operatorname{arsech}\,x)' = -{1 \over x\sqrt{1 - x^2}}$
$(\operatorname{csch}\,x)' = -\,\operatorname{coth}\,x\,\operatorname{csch}\,x$	$(\operatorname{arcsch}\,x)' = -{1 \over |x|\sqrt{1 + x^2}}$
$(\operatorname{coth}\,x )' = -\,\operatorname{csch}^2\,x$	$(\operatorname{arcoth}\,x)' = { 1 \over 1 - x^2}$

Derivatives of special functions

Gamma function $(\Gamma(x))' = \int_0^\infty t^{x-1} e^{-t} \ln t\,dt$ $(\Gamma(x))' = \Gamma(x) \left(\sum_{n=1}^\infty \left(\ln\left(1 + \dfrac{1}{n}\right) - \dfrac{1}{x + n}\right) - \dfrac{1}{x}\right) = \Gamma(x) \psi(x)$

Riemann Zeta function $(\zeta(x))' = -\sum_{n=1}^\infty \frac{\ln n}{n^x} = -\frac{\ln 2}{2^x} - \frac{\ln 3}{3^x} - \frac{\ln 4}{4^x} - \cdots \!$ $(\zeta(x))' = -\sum_{p \text{ prime}} \frac{p^{-x} \ln p}{(1-p^{-x})^2}\prod_{q \text{ prime}, q \neq p} \frac{1}{1-q^{-x}} \!$

Nth Derivatives

The following formulae can be obtained empirically by repeated differentiation and taking notice of patterns; either by hand or computed by a CAS (Computer Algebra System).^[1] Below y is the dependent variable, x is the independent variable, real number constants are A, B, N, r, real integers are n and j, F(x) is a continuously differentiable function (the n^th derivative exists), and i is the imaginary unit $\sqrt{-1} \!$.

Function	nth Derivative
$y = F(G(x)) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = n! \displaystyle\sum_{\{k_m\}}^{} \dfrac{\mathrm{d}^r}{\mathrm{d} z^r} F(z)|_{z=G(x)} \displaystyle\prod_{m=1}^n \dfrac{1}{k_m!} \left(\dfrac{1}{m!} \dfrac{\mathrm{d}^m}{\mathrm{d} x^m} G(x) \right)^{k_m} \!$ where $r = \displaystyle\sum_{m=1}^{n-1} k_m \!$ and the set $\{k_m\} \!$ consists of all non-negative integer solutions of the Diophantine equation $\displaystyle\sum_{m=1}^{n} m k_m = n \!$ See: Faà di Bruno's formula, Expansions for nearly Gaussian distributions by S. Blinnikov and R. Moessner [2]
$y = F(x)G(x) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = \displaystyle\sum_{k=0}^{n} \displaystyle\binom{n}{k} \dfrac{\mathrm{d}^{n-k}}{\mathrm{d} x^{n-k}} F(x) \dfrac{\mathrm{d}^k}{\mathrm{d} x^k} G(x) \!$ See: General Leibniz rule
$y = x^N \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = \displaystyle\prod_{r=1}^n (N-r+1)x^{N-n} \!$
$y = [F(x)]^r \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = r \displaystyle\binom{n-r}{n} \displaystyle\sum_{j=0}^{n} \dfrac{(-1)^j}{r-j}{\displaystyle\binom{n}{j} [F(x)]^{r-j} \dfrac{\mathrm{d}^n}{\mathrm{d} x^n} [F(x)]^j} \!$
$y = B^{Ax} \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n B^{Ax} \left ( \ln{B} \right )^n \!$
For the case of $B = \exp(1) = e \!$ (the exponential function), the above reduces to: $y = e^{Ax} \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n e^{Ax} \!$
$y = \ln[F(x)] \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = \delta_n \ln[F(x)] + \displaystyle\sum_{j=1}^n \dfrac{(-1)^{j-1}}{j} \binom{n}{j} \dfrac{1}{[F(x)]^j} \dfrac{\mathrm{d}^n}{\mathrm{d} x^n} [F(x)]^j \!$ where $\delta_n = \begin{cases} 1 & n=0 \\ 0 & n \neq 0 \\ \end{cases} \!$ is the Kronecker delta.
$y = \sin(A x + B) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n \sin \left ( A x + B + \frac{n \pi}{2} \right ) \!$ Expanding this by the sine addition formula yields a more clear form to use: $\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n \left [ \cos \left (\dfrac{n \pi}{2} \right ) \sin \left ( A x + B \right ) + \sin \left ( \dfrac{n \pi}{2} \right ) \cos \left ( A x + B \right ) \right ] \!$
$y = \cos(A x + B) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n \cos \left ( A x + B + \frac{n \pi}{2} \right ) \!$ Expanding by the cosine addition formula: $\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = A^n \left [ \cos \left (\dfrac{n \pi}{2} \right ) \cos \left ( A x + B \right ) - \sin \left ( \dfrac{n \pi}{2} \right ) \sin \left ( A x + B \right ) \right ] \!$
$y = \sinh(A x + B) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = (-i A^n) \sinh \left ( Ax + B + \dfrac{ i n \pi}{2} \right ) \!$
$y = \cosh(A x + B) \!$	$\dfrac{\mathrm{d}^n y}{\mathrm{d} x^n} = (\pm i A^n) \cosh \left ( Ax + B \mp \dfrac{ i n \pi}{2} \right ) \!$

See also

• Derivative

References

1. ^ http://www.wolframalpha.com/input/?i=nth+derivative+of+%5Bf%28x%29%5D%5Er
2. ^ http://aas.aanda.org/index.php?option=com_article&access=standard&Itemid=129&url=/articles/aas/pdf/1998/10/h0596.pdf

External Links

Derivative calculator with formula simplification

• A Table of Derivatives