Differentiation rules

Unless otherwise stated, all functions are functions of real numbers ($\backslash mathbb\{R\}$) that return real values, although, more generally, the formulas below apply wherever they are well defined,Calculus (5th edition), F. Ayres, E. Mendelson, Schaum's Outline Series, 2009, {{ISBN|978-0-07-150861-2}}.Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, {{ISBN|978-0-07-162366-7}}. including the case of complex numbers ($\backslash mathbb\{C\}$).Complex Variables, M.R. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009, {{ISBN|978-0-07-161569-3}}

For any value of $c$, where $c\; \backslash in\; \backslash mathbb\{R\}$, if $f(x)$ is the constant function given by $f(x)\; =\; c$, then $\backslash frac\{df\}\{dx\}\; =\; 0$.{{cite web |title=Differentiation Rules |url=https://courseware.cemc.uwaterloo.ca/11/assignments/47/6 |website=University of Waterloo – CEMC Open Courseware |access-date=3 May 2022}}

Let $c\; \backslash in\; \backslash mathbb\{R\}$ and $f(x)\; =\; c$. By the definition of the derivative:

$$\backslash begin\{align\}$$

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}

This computation shows that the derivative of any constant function is 0.

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.

In other words, the value of the constant function, $y$, will not change as the value of $x$ increases or decreases.

File:Tangent function animation.gif 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 0 where black and solid.]]

{{Main|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 $h\text{'}(x)\; =\; a\; f\text{'}(x)\; +\; b\; g\text{'}(x)$.

In Leibniz's notation, this formula is written as:

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

Special cases include:

• The constant factor rule:

$$(af)\text{'}\; =\; af\text{'},$$

• The sum rule:

$$(f\; +\; g)\text{'}\; =\; f\text{'}\; +\; g\text{'},$$

• The difference rule:

$$(f\; -\; g)\text{'}\; =\; f\text{'}\; -\; g\text{'}.$$

{{Main|Product rule}}

For the functions $f$ and $g$, the derivative of the function $h(x)\; =\; f(x)\; g(x)$ with respect to $x$ is:

$$h\text{'}(x)\; =\; (fg)\text{'}(x)\; =\; f\text{'}(x)\; g(x)\; +\; f(x)\; g\text{'}(x).$$

In Leibniz's notation, this formula is written:

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

{{Main|Chain rule}}

The derivative of the function $h(x)\; =\; f(g(x))$ is:

$$h\text{'}(x)\; =\; f\text{'}(g(x))\backslash cdot\; g\text{'}(x).$$

In Leibniz's notation, this formula is written as:

$$\backslash frac\{d\}\{dx\}h(x)\; =\; \backslash left.\backslash frac\{d\}\{dz\}f(z)\backslash right|\_\{z=g(x)\}\backslash cdot\; \backslash frac\{d\}\{dx\}g(x),$$

often abridged to:

$$\backslash frac\{dh(x)\}\{dx\}\; =\; \backslash frac\{df(g(x))\}\{dg(x)\}\; \backslash cdot\; \backslash frac\{dg(x)\}\{dx\}.$$

Focusing on the notion of maps, and the differential being a map $\backslash text\{D\}$, this formula is written in a more concise way as:

$$[\backslash text\{D\}\; (f\backslash circ\; g)]\_x\; =\; [\backslash text\{D\}\; f]\_\{g(x)\}\; \backslash cdot\; [\backslash text\{D\}g]\_x.$$

{{Main|Inverse function rule}}

If the function $f$ has an inverse function $g$, meaning that $g(f(x))\; =\; x$ and $f(g(y))\; =\; y$, then:

$$g\text{'}\; =\; \backslash frac\{1\}\{f\text{'}\backslash circ\; g\}.$$

In Leibniz notation, this formula is written as:

$$\backslash frac\{dx\}\{dy\}\; =\; \backslash frac\{1\}\{\backslash frac\{dy\}\{dx\}\}.$$

{{Main|Power rule}}

If $f(x)\; =\; x^r$, for any real number $r\; \backslash neq\; 0$, then:

$$f\text{'}(x)\; =\; rx^\{r-1\}.$$

When $r\; =\; 1$, this formula becomes the special case that, if $f(x)\; =\; x$, then $f\text{'}(x)\; =\; 1$.

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

{{Main|Reciprocal rule}}

The derivative of $h(x)=\backslash frac\{1\}\{f(x)\}$ for any (nonvanishing) function $f$ is:

$$h\text{'}(x)\; =\; -\backslash frac\{f\text{'}(x)\}\{(f(x))^2\},$$

wherever $f$ is nonzero.

In Leibniz's notation, this formula is written:

$$\backslash frac\{d\backslash left(\backslash frac\{1\}\{f\}\backslash right)\}\{dx\}\; =\; -\backslash frac\{1\}\{f^2\}\backslash frac\{df\}\{dx\}.$$

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

{{Main|Quotient rule}}

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

$$\backslash left(\backslash frac\{f\}\{g\}\backslash right)\text{'}\; =\; \backslash frac\{f\text{'}g\; -\; g\text{'}f\}\{g^2\},$$

wherever $g$ is nonzero.

This can be derived from the product rule and the reciprocal rule.

{{Main|Power rule}}

The elementary power rule generalizes considerably. The most general power rule is the functional power rule: for any functions $f$ and $g$,

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

wherever both sides are well defined.

Special cases:

• If $f(x)=x^a$, then $f\text{'}(x)=ax^\{a-1\}$ when $a$ is any nonzero real number and $x$ is positive.
• The reciprocal rule may be derived as the special case where $g(x)=-1\backslash !$.

$$\backslash frac\{d\}\{dx\}\backslash left(c^\{ax\}\backslash right)\; =\; \{ac^\{ax\}\; \backslash ln\; c\; \}\; ,\backslash qquad\; c\; >\; 0.$$

The equation above is true for all $c$, but the derivative for yields a complex number.

$$\backslash frac\{d\}\{dx\}\backslash left(e^\{ax\}\backslash right)\; =\; ae^\{ax\}.$$

$$\backslash frac\{d\}\{dx\}\backslash left(\; \backslash log\_c\; x\backslash right)\; =\; \{1\; \backslash over\; x\; \backslash ln\; c\}\; ,\; \backslash qquad\; c\; >\; 1.$$

The equation above is also true for all $c$ but yields a complex number if .

$$\backslash frac\{d\}\{dx\}\backslash left(\; \backslash ln\; x\backslash right)\; =\; \{1\; \backslash over\; x\}\; ,\backslash qquad\; x\; >\; 0.$$

$$\backslash frac\{d\}\{dx\}\backslash left(\; \backslash ln\; |x|\backslash right)\; =\; \{1\; \backslash over\; x\}\; ,\backslash qquad\; x\; \backslash neq\; 0.$$

$$\backslash frac\{d\}\{dx\}\backslash left(\; W(x)\backslash right)\; =\; \{1\; \backslash over\; \{x+e^\{W(x)\}\}\}\; ,\backslash qquad\; x\; >\; -\{1\; \backslash over\; e\},$$

where $W(x)$ is the Lambert W function.

$$\backslash frac\{d\}\{dx\}\backslash left(\; x^x\; \backslash right)\; =\; x^x(1+\backslash ln\; x).$$

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

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

$$(\backslash ln\; f)\text{'}=\; \backslash frac\{f\text{'}\}\{f\},$$

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|date=October 2021}}

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.

{{Main|Differentiation of trigonometric functions}}

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

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

$$\backslash frac\{\backslash partial\; \backslash arctan(y,x)\}\{\backslash partial\; y\}\; =\; \backslash frac\{x\}\{x^2\; +\; y^2\}\; \backslash qquad\backslash text\{and\}\backslash qquad\; \backslash frac\{\backslash partial\; \backslash arctan(y,x)\}\{\backslash partial\; x\}\; =\; \backslash frac\{-y\}\{x^2\; +\; y^2\}.$$

{{Main|Hyperbolic functions#Derivatives}}

{{Main|Gamma function}}

$$\backslash Gamma(x)\; =\; \backslash int\_0^\backslash infty\; t^\{x-1\}\; e^\{-t\}\backslash ,\; dt$$

$$\backslash begin\{align\}$$

\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}

with $\backslash psi(x)$ being the digamma function, expressed by the parenthesized expression to the right of $\backslash Gamma(x)$ in the line above.

{{Main|Riemann zeta function}}

$$\backslash zeta(x)\; =\; \backslash sum\_\{n=1\}^\backslash infty\; \backslash frac\{1\}\{n^x\}$$

$$\backslash begin\{align\}$$

\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}

{{Main|Differentiation under the integral sign}}

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

$$F(x)=\backslash int\_\{a(x)\}^\{b(x)\}f(x,t)\backslash ,dt,$$

where the functions $f(x,t)$ and $\backslash frac\{\backslash partial\}\{\backslash partial\; x\}\backslash ,f(x,t)$ are both continuous in both $t$ and $x$ in some region of the $(t,x)$ plane, including $a(x)\backslash leq\; t\backslash leq\; b(x)$, where $x\_0\backslash leq\; x\backslash leq\; x\_1$, and the functions $a(x)$ and $b(x)$ are both continuous and both have continuous derivatives for $x\_0\backslash leq\; x\backslash leq\; x\_1$. Then, for $\backslash ,x\_0\backslash leq\; x\backslash leq\; x\_1$:

$$F\text{'}(x)\; =\; f(x,b(x))\backslash ,b\text{'}(x)\; -\; f(x,a(x))\backslash ,a\text{'}(x)\; +\; \backslash int\_\{a(x)\}^\{b(x)\}\; \backslash frac\{\backslash partial\}\{\backslash partial\; x\}\backslash ,\; f(x,t)\backslash ;\; dt\backslash ,.$$

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, including:

{{Main|Faà di Bruno's formula}}

If $f$ and $g$ are $n$-times differentiable, then:

$$\backslash frac\{d^n\}\{d\; x^n\}\; [f(g(x))]=\; n!\; \backslash sum\_\{\backslash \{k\_m\backslash \}\}\; f^\{(r)\}(g(x))\; \backslash prod\_\{m=1\}^n\; \backslash frac\{1\}\{k\_m!\}\; \backslash left(g^\{(m)\}(x)\; \backslash right)^\{k\_m\},$$

where $r\; =\; \backslash sum\_\{m=1\}^\{n-1\}\; k\_m$ and the set $\backslash \{k\_m\backslash \}$ consists of all non-negative integer solutions of the Diophantine equation $\backslash sum\_\{m=1\}^\{n\}\; m\; k\_m\; =\; n$.

{{Main|General Leibniz rule}}

If $f$ and $g$ are $n$-times differentiable, then:

$$\backslash frac\{d^n\}\{dx^n\}[f(x)g(x)]\; =\; \backslash sum\_\{k=0\}^\{n\}\; \backslash binom\{n\}\{k\}\; \backslash frac\{d^\{n-k\}\}\{d\; x^\{n-k\}\}\; f(x)\; \backslash frac\{d^k\}\{d\; x^k\}\; g(x).$$

• {{annotated link|Differentiable function}}
• {{annotated link|Differential of a function}}
• {{annotated link|Differentiation of integrals}}
• {{annotated link|Differentiation under the integral sign}}
• {{annotated link|Hyperbolic functions}}
• {{annotated link|Inverse hyperbolic functions}}
• {{annotated link|Inverse trigonometric functions}}
• {{annotated link|Lists of integrals}}
• {{annotated link|List of mathematical functions}}
• {{annotated link|Matrix calculus}}
• {{annotated link|Trigonometric functions}}
• {{annotated link|Vector calculus identities}}

{{reflist}}

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, {{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}}.

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• [http://www.planetcalc.com/675/ Derivative calculator with formula simplification]
• [https://semanticdirectory.com.ua/math?lang=en&#TableOfDerivatives The table of derivatives with animated proves]

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