Editing Derivative (section)

== Rules of computation ==
{{Main|Differentiation rules}}
In principle, the derivative of a function can be computed from the definition by considering the difference quotient and computing its limit. Once the derivatives of a few simple functions are known, the derivatives of other functions are more easily computed using ''rules'' for obtaining derivatives of more complicated functions from simpler ones. This process of finding a derivative is known as '''differentiation'''.{{sfn|Apostol|1967|p=160}}

=== Rules for basic functions ===
The following are the rules for the derivatives of the most common basic functions. Here, <math> a </math> is a real number, and <math> e </math> is [[e (mathematical constant)|the base of the natural logarithm, approximately {{nowrap|1=2.71828}}]].<ref>{{harvnb|Varberg|Purcell|Rigdon|2007}}. See p. 133 for the power rule, pp. 115–116 for the trigonometric functions, p. 326 for the natural logarithm, pp. 338–339 for exponential with base {{tmath|1= e }}, p. 343 for the exponential with base {{tmath|1= a }}, p. 344 for the logarithm with base {{tmath|1= a }}, and p. 369 for the inverse of trigonometric functions.</ref> 
* ''[[Power rule|Derivatives of powers]]'':
*: <math> \frac{d}{dx}x^a = ax^{a-1} </math>
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* ''Functions of [[Exponential function|exponential]], [[natural logarithm]], and [[logarithm]] with general base'':
*: <math> \frac{d}{dx}e^x = e^x </math>
*: <math> \frac{d}{dx}a^x = a^x\ln(a)  </math>, for <math> a > 0 </math>
*: <math> \frac{d}{dx}\ln(x) = \frac{1}{x} </math>, for <math> x > 0 </math>
*: <math> \frac{d}{dx}\log_a(x) = \frac{1}{x\ln(a)}  </math>, for <math> x, a > 0 </math>
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* ''[[Trigonometric functions]]'':
*: <math> \frac{d}{dx}\sin(x) = \cos(x) </math>
*: <math> \frac{d}{dx}\cos(x) = -\sin(x) </math>
*: <math> \frac{d}{dx}\tan(x) = \sec^2(x) = \frac{1}{\cos^2(x)} = 1 + \tan^2(x) </math>
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* ''[[Inverse trigonometric functions]]'':
*: <math> \frac{d}{dx}\arcsin(x) = \frac{1}{\sqrt{1-x^2}} </math>, for <math> -1 < x < 1 </math>
*: <math> \frac{d}{dx}\arccos(x)= -\frac{1}{\sqrt{1-x^2}} </math>, for <math> -1 < x < 1 </math>
*: <math> \frac{d}{dx}\arctan(x)= \frac{1}{{1+x^2}} </math>
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=== Rules for combined functions <span class="anchor" id="Rules"></span> ===
Given that the <math> f </math> and <math> g </math> are the functions. The following are some of the most basic rules for deducing the derivative of functions from derivatives of basic functions.<ref> For constant rule and sum rule, see {{harvnb|Apostol|1967|pp=161, 164}}, respectively. For the product rule, quotient rule, and chain rule, see {{harvnb|Varberg|Purcell|Rigdon|2007|pp=111–112, 119}}, respectively. For the special case of the product rule, that is, the product of a constant and a function, see {{harvnb|Varberg|Purcell|Rigdon|2007|pp=108–109}}.</ref>
* ''Constant rule'': if <math>f</math> is constant, then for all {{tmath|1= x }},
*: <math>f'(x) = 0. </math>
* ''[[Linearity of differentiation|Sum rule]]'':
*: <math>(\alpha f + \beta g)' = \alpha f' + \beta g' </math> for all functions <math>f</math> and <math>g</math> and all real numbers <math>\alpha</math> and {{tmath|1= \beta}}.
* ''[[Product rule]]'':
*: <math>(fg)' = f 'g + fg' </math> for all functions <math>f</math> and {{tmath|1= g }}. As a special case, this rule includes the fact <math>(\alpha f)' = \alpha f'</math> whenever <math>\alpha</math> is a constant because <math>\alpha' f = 0 \cdot f = 0</math> by the constant rule.
* ''[[Quotient rule]]'':
*: <math>\left(\frac{f}{g} \right)' = \frac{f'g - fg'}{g^2}</math>  for all functions <math>f</math> and <math>g</math> at all inputs where {{nowrap|''g'' ≠ 0}}.
* ''[[Chain rule]]'' for [[Function composition|composite functions]]: If {{tmath|1= f(x) = h(g(x)) }}, then
*: <math>f'(x) = h'(g(x)) \cdot g'(x). </math>

=== Computation example ===
The derivative of the function given by <math>f(x) = x^4 + \sin \left(x^2\right) - \ln(x) e^x + 7</math> is
<math display="block"> \begin{align}
 f'(x) &= 4 x^{(4-1)}+ \frac{d\left(x^2\right)}{dx}\cos \left(x^2\right) - \frac{d\left(\ln {x}\right)}{dx} e^x - \ln(x) \frac{d\left(e^x\right)}{dx} + 0 \\
 &= 4x^3 + 2x\cos \left(x^2\right) - \frac{1}{x} e^x - \ln(x) e^x.
\end{align} </math>
Here the second term was computed using the [[chain rule]] and the third term using the [[product rule]]. The known derivatives of the elementary functions <math> x^2 </math>, <math> x^4 </math>, <math> \sin (x) </math>, <math> \ln (x) </math>, and <math> \exp(x) = e^x </math>, as well as the constant <math> 7 </math>, were also used.