Differentiation rules explained

This article 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 ($\backslash mathbb$) that return real values, although, more generally, the formulas below apply wherever they are well defined,^{[1] [2]} including the case of complex numbers ($\backslash mathbb$).^[3]

Constant term rule

For any value of $c$, where $c\; \backslash in\; \backslash mathbb$, if $f(x)$ is the constant function given by $f(x)\; =\; c$, then $\backslash frac\; =\; 0$.^[4]

Proof

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

This computation 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. 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.

Differentiation is linear

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

In Leibniz's notation, this formula is written as:$$\backslash frac\; =\; a\backslash frac\; +b\backslash frac.$$

Special cases include:

• The constant factor rule:

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

• The sum rule:

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

• The difference rule:

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

Product rule

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

In Leibniz's notation, this formula is written:$$\backslash frac\; =\; g\; \backslash frac\; +\; f\; \backslash frac.$$

Chain rule

See main article: 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:$$\backslash frach(x)\; =\; \backslash left.\backslash fracf(z)\backslash right|\_\backslash cdot\; \backslash fracg(x),$$often abridged to:$$\backslash frac\; =\; \backslash frac\; \backslash cdot\; \backslash frac.$$

Focusing on the notion of maps, and the differential being a map $\backslash text$, this formula is written in a more concise way as:$$[\backslash text\{D\}\; (f\backslash circ\; g)]\_x\; =\; [\backslash text\{D\}\; f]\_\; \backslash cdot\; [\backslash text\{D\}g]\_x.$$

Inverse function rule

See main article: Inverse function rule. If the function $f$ has an inverse function $g$, meaning that $g(f(x))\; =\; x$ and $f(g(y))\; =\; y$, then:$$g\text{'}\; =\; \backslash frac.$$

In Leibniz notation, this formula is written as:$$\backslash frac\; =\; \backslash frac.$$

Power laws, polynomials, quotients, and reciprocals

Polynomial or elementary power rule

See main article: Power rule. If $f(x)\; =\; x^r$, for any real number $r\; \backslash neq\; 0$, then:$$f\text{'}(x)\; =\; rx^.$$

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.

Reciprocal rule

See main article: Reciprocal rule. The derivative of $h(x)=\backslash frac$ for any (nonvanishing) function $f$ is:$$h\text{'}(x)\; =\; -\backslash frac,$$wherever $f$ is nonzero.

In Leibniz's notation, this formula is written:$$\backslash frac\; =\; -\backslash frac\backslash frac.$$

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

Quotient rule

See main article: Quotient rule. If $f$ and $g$ are functions, then:$$\backslash left(\backslash frac\backslash right)\text{'}\; =\; \backslash frac,$$wherever $g$ is nonzero.

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

Generalized power rule

See 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 $g$,$$(f^g)\text{'}\; =\; \backslash left(e^\backslash right)\text{'}\; =\; f^g\backslash left(f\text{'}\; +\; g\text{'}\backslash ln\; f\backslash right),\backslash quad$$wherever both sides are well defined.

Special cases:

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

Derivatives of exponential and logarithmic functions

$$\backslash frac\backslash left(c^\backslash right)\; =,\backslash qquad\; c\; >\; 0.$$The equation above is true for all

c , but the derivative for c< 0 yields a complex number.

$$\backslash frac\backslash left(e^\backslash right)\; =\; ae^.$$

$$\backslash frac\backslash left(\backslash log\_c\; x\backslash right)\; =,\; \backslash qquad\; c\; >\; 1.$$The equation above is also true for all $c$ but yields a complex number if .

$$\backslash frac\backslash left(\backslash ln\; x\backslash right)\; =,\backslash qquad\; x\; >\; 0.$$

$$\backslash frac\backslash left(\backslash ln\; |x|\backslash right)\; =,\backslash qquad\; x\; \backslash neq\; 0.$$

$$\backslash frac\backslash left(W(x)\backslash right)\; =,\backslash qquad\; x\; >\; -,$$where $W(x)$ is the Lambert W function.

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

$$\backslash frac\backslash left(f(x)^\; \backslash right)\; =\; g(x)f(x)^\; \backslash frac\; +\; f(x)^\backslash ln\backslash frac,\; \backslash qquad\; \backslash textf(x)\; >\; 0\; \backslash text\; \backslash frac\; \backslash text\; \backslash frac\; \backslash text$$

$$\backslash frac\backslash left(f\_(x)^\; \backslash right)\; =\; \backslash left\; [\backslash sum\backslash limits\_\{k=1\}^\{n\}\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; x\_\{k\}\}\; \backslash left(f\_\{1\}(x\_1)^\{f\_\{2\}(x\_2)^\{\backslash left\; (...\; \backslash right)^\{f\_\{n\}(x\_n)\}\}\}\; \backslash right)\; \backslash right\; ]\; \backslash biggr\backslash vert\_,\backslash qquad\; \backslash text\; f\_(x)\; >\; 0\; \backslash text\backslash frac\; \backslash text$$

Logarithmic derivatives

The logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule):$$(\backslash ln\; f)\text{'}=\; \backslash frac,$$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.

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

See main article: Differentiation of trigonometric functions.

width=50%	(d)/(dx) \sin x = \cos x	width=50%	(d)/(dx) \arcsin x = (1)/(\sqrt{ 1- x ^{2}) }
(d)/(dx) \cos x = - \sin x	(d)/(dx) \arccos x =- (1)/(\sqrt{ 1- x ^{2}) }
(d)/(dx) \tan x = \sec ^{2} x = (1)/(\cos ^{2} x) = 1 + \tan ^{2} x	(d)/(dx) \arctan x = (1)/(1+ x ^{2})
(d)/(dx) \csc x = - \csc{x}\cot{x}	(d)/(dx) \operatorname{arccsc} x =- (1)/(|x|\sqrt{x ^{2} -1) }
(d)/(dx) \sec x = \sec{x}\tan{x}	(d)/(dx) \operatorname{arcsec} x = (1)/(|x|\sqrt{x ^{2} -1) }
(d)/(dx) \cot x = - \csc ^{2} x =- (1)/(\sin ^{2} x) = -1 - \cot ^{2} x	(d)/(dx) \operatorname{arccot} x = - { 1 \over 1 + x ^{2}}

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 $\backslash arctan(y,\; x>0)\; =\; \backslash arctan(\backslash frac)$. Its partial derivatives are:$$\backslash frac\; =\; \backslash frac\; \backslash qquad\backslash text\backslash qquad\; \backslash frac\; =\; \backslash frac.$$

Derivatives of hyperbolic functions

width=50%	(d)/(dx) \sinh x = \cosh x	(d)/(dx) \operatorname{arsinh}x = (1)/(\sqrt{ 1+ x ^{2}) }
(d)/(dx) \cosh x = \sinh x	(d)/(dx) \operatorname{arcosh} x = { (1)/(\sqrt{x ^{2-1}) }}
(d)/(dx) \tanh x = {\operatorname{sech} ^{2} x} = 1 - \tanh ^{2} x	(d)/(dx) \operatorname{artanh} x = (1)/(1- x ^{2})
(d)/(dx) \operatorname{csch} x = - \operatorname{csch}{x}\coth{x}	(d)/(dx) \operatorname{arcsch} x =- (1)/(|x|\sqrt{ 1+ x ^{2}) }
(d)/(dx) \operatorname{sech} x = - \operatorname{sech}{x}\tanh{x}	(d)/(dx) \operatorname{arsech} x =- (1)/(x\sqrt{ 1- x ^{2}) }
(d)/(dx) \coth x = - \operatorname{csch} ^{2} x = 1 - \coth ^{2} x	(d)/(dx) \operatorname{arcoth} x = (1)/(1- x ^{2})

Derivatives of special functions

Gamma function

See main article: Gamma function. $$\backslash Gamma(x)\; =\; \backslash int\_0^\backslash infty\; t^\; e^\backslash ,\; dt$$with $\backslash psi(x)$ being the digamma function, expressed by the parenthesized expression to the right of $\backslash Gamma(x)$ in the line above.

Riemann zeta function

See main article: Riemann zeta function. $$\backslash zeta(x)\; =\; \backslash sum\_^\backslash infty\; \backslash frac$$

Derivatives of integrals

Suppose that it is required to differentiate with respect to $x$ the function:$$F(x)=\backslash int\_^f(x,t)\backslash ,dt,$$

where the functions $f(x,t)$ and $\backslash frac\backslash ,f(x,t)$ 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 $\backslash ,x\_0\backslash leq\; x\backslash leq\; x\_1$:$$F\text{'}(x)\; =\; f(x,b(x))\backslash ,b\text{'}(x)\; -\; f(x,a(x))\backslash ,a\text{'}(x)\; +\; \backslash int\_^\; \backslash frac\backslash ,\; f(x,t)\backslash ;\; dt\backslash ,.$$

This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus.

Derivatives to nth order

Some rules exist for computing the $n$th derivative of functions, where $n$ is a positive integer, including:

Faà di Bruno's formula

See main article: Faà di Bruno's formula. If $f$ and $g$ are $n$-times differentiable, then:$$\backslash frac\; [f(g(x))]=\; n!\; \backslash sum\_\; f^(g(x))\; \backslash prod\_^n\; \backslash frac\; \backslash left(g^(x)\; \backslash right)^,$$where $r\; =\; \backslash sum\_^\; k\_m$ and the set $\backslash$ consists of all non-negative integer solutions of the Diophantine equation $\backslash sum\_^\; m\; k\_m\; =\; n$.

General Leibniz rule

See main article: General Leibniz rule. If $f$ and $g$ are $n$-times differentiable, then:$$\backslash frac[f(x)g(x)]\; =\; \backslash sum\_^\; \backslash binom\; \backslash frac\; f(x)\; \backslash frac\; g(x).$$

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

External links

• Derivative calculator with formula simplification

Notes and References

1. Calculus (5th edition), F. Ayres, E. Mendelson, Schaum's Outline Series, 2009, .
2. Advanced Calculus (3rd edition), R. Wrede, M.R. Spiegel, Schaum's Outline Series, 2010, .
3. Complex Variables, M.R. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outlines Series, McGraw Hill (USA), 2009,
4. Web site: Differentiation Rules . University of Waterloo – CEMC Open Courseware . 3 May 2022.