Factsheet: Rules of calculus
Please note: clickable links lead to study guides where the rule is introduced.
Rules of differentiation
Limit definition of the derivative: If \(f(x)\) is a continuous function, then (if it exists) the derivative \(f'(x)\) is defined by \[\lim_{h\to 0}\frac{f(x+h) - f(x)}{h}\]
Sum/difference and constant rule: If \(f(x)\) and \(g(x)\) are differentiable functions, then \[\frac{\textrm{d}}{\textrm{d}x}(f(x) \pm g(x)) = f'(x) \pm g'(x)\quad \textsf{ and }\quad \frac{\textrm{d}}{\textrm{d}x}(cf(x)) = c\frac{\textrm{d}}{\textrm{d}x}(f(x)) = cf'(x)\]
Product rule: If \(f(x) = u(x)v(x)\), \[f'(x) = \frac{\textrm{d}}{\textrm{d}x}\left(u(x)v(x)\right) = u(x)v'(x) + u'(x)v(x)\]
Quotient rule: If \(f(x) = u(x)/v(x)\) and \(v(x) \neq 0\), then \[f'(x) = \frac{\textrm{d}}{\textrm{d}x}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x)u'(x) - u(x)v'(x)}{(v(x))^2}\]
Chain rule: If \(f(x) = f(u(x))\), then \[f'(x) = \frac{\textrm{d}f}{\textrm{d}u}\cdot \frac{\textrm{d}u}{\textrm{d}x} = f'(u(x))\cdot u'(x)\] where \(f'(u(x))\) is the derivative of \(f(u)\) with respect to \(u\).
Implicit differentiation: If \(f(x,y) = 0\) defines a function \(g(y)\) implicitly, then \[\frac{\textrm{d}}{\textrm{d}x}(g(y)) = \frac{\textrm{d}g}{\textrm{d}y}\cdot\frac{\textrm{d}y}{\textrm{d}x} = g'(y)\cdot \frac{\textrm{d}y}{\textrm{d}x}\] where \(g'(y)\) is the derivative of \(g(y)\) with respect to \(y\).
Rules of integration
Sum/difference and constant rules: If \(f,g\) are functions and \(k\) is any number: \[\int f(x)\pm g(x) \,\mathrm{d}x = \int f(x) \,\mathrm{d}x \pm \int g(x) \,\mathrm{d}x \quad\textsf{ and }\quad \int kf(x)\,\textrm{d}x = k\int f(x)\,\textrm{d}x\]
Limit manipulation: If \(f\) is a function and \(a,b\) are real numbers, then:
for \(c\) such that \(a < c <b\), then: \[\int_a^bf(x)\,\mathrm{d}x = \int_a^c {f(x)}\,\mathrm{d}x + \int_c^b {f(x)}\,\mathrm{d}x\]
if \(a \leq b\), then: \[\int_a^b f(x)\,\mathrm{d}x = -\int_b^a {f(x)}\,\mathrm{d}x\]
Integration by substitution: For an indefinite integral, \[\int f(u(x))\cdot u'(x) \,\textrm{d}x = \int f(u)\,\textrm{d}u\] and for a definite integral \[\int_a^b f(u(x))\cdot u'(x) \,\textrm{d}x = \int_{u(a)}^{u(b)} f(u)\,\textrm{d}u\]
Integration by parts: For functions \(u,v\) of \(x\): \[\int uv'\textrm{d}x = uv - \int {vu'}\,\textrm{d}x\]
Integration of derivative over function: For a function \(f\), \[\int \frac{f'(x)}{f(x)} \,\textrm{d}x = \ln|f(x)| + C.\]
Version history
v1.0: created in 08/25 by tdhc.