Derivatives of common elementary functions table
Table shows derivatives of selected elementary functions of one variable f(x) such as linear function, square function, sine, cosine, logarithm etc.

Derivative formulas#

Function f(x)Derivative f'(x)Notice
Show sourceaaShow source00Show source
Show sourcexxShow source11Show source
Show sourceax+bax+bShow sourceaaShow source
Show sourceax2+bx+cax^2+bx+cShow source2ax+b2ax+bShow source
Show sourcexax^aShow sourceaxa1ax^{a-1}Show source
Show sourcex\sqrt{x}Show source12x\frac{1}{2\sqrt{x}}Show source
Show sourcexn\sqrt[n]{x}Show source1nxn1n\frac{1}{n\sqrt[n]{x^{n-1}}}Show sourcenN\{0,1}n \in N \backslash \{0,1\}
Show source1x\frac{1}{x}Show source1x2\frac{-1}{x^2}Show source
Show sourceax\frac{a}{x}Show sourceax2\frac{-a}{x^2}Show source
Show sourcesinx\sin xShow sourcecosx\cos xShow source
Show sourcecosx\cos xShow sourcesinx-\sin xShow source
Show sourcetgx\tg xShow source1cos2x\frac{1}{\cos^2 x}Show source
Show sourcectgx\ctg xShow source1sin2x-\frac{1}{\sin^2 x}Show source
Show sourceaxa^xShow sourceaxlnaa^x \cdot \ln aShow source
Show sourceexe^xShow sourceexe^xShow source
Show sourcelnx\ln xShow source1x\frac{1}{x}Show source
Show sourcelnx\ln|x|Show source1x\frac{1}{x}Show source
Show sourcelogax\log_axShow source1xlna\frac{1}{x \ln a}Show source
Show sourcearcsinxarc \sin xShow source11x2\frac{1}{\sqrt{1 - x^2}}Show source
Show sourcearccosxarc \cos xShow source11x2\frac{-1}{\sqrt{1 - x^2}}Show source
Show sourcearctgxarc \tg xShow source11+x2\frac{1}{1 + x^2}Show source
Show sourcearcctgxarc \ctg xShow source11+x2\frac{-1}{1 + x^2}Show source

Some facts#

  • The derivative in point is defined as the limit of so-called difference quotient with x going to this point:
    f(x0)=limxx0f(x)f(x0)xx0f'(x_0) = \lim\limits_{x \to x_0} \dfrac{f(x) - f(x_0)}{x - x_0}
  • Above formula is often given in alternative form (which is sometimes more useful depending on context). In this form we assume x=x0+hx = x_0 + h, where hh is so-called "very small change" of x variable (independent variable):
    f(x)=limh0f(x+h)f(x)hf'(x) = \lim\limits_{h \to 0} \dfrac{f(x + h) - f(x)}{h}
  • Derivative understood as a function, assigns the above formula (i.e. limit of difference quotient) to each point from domain.
    f:xlimh0f(x+h)f(x)hf': x \rightarrow \lim\limits_{h \to 0} \dfrac{f(x + h) - f(x)}{h}
  • ⓘ Example: The derivative of linear function f(x)=ax+bf(x) = ax + b, is f'(x) = a, because:
    (ax+b)=deflimh0a(x+h)+b(ax+b)h=limh0ax+ah+baxbh=limh0ahh=a(ax+b)' \overset{\mathrm{def}}{=} \lim\limits_{h \to 0} \dfrac{a \cdot (x + h) + b - (ax + b)}{h} = \lim\limits_{h \to 0} \dfrac{\cancel{ax} + ah + \cancel{b} - \cancel{ax} - \cancel{b}}{h} = \lim\limits_{h \to 0} \dfrac{a\cancel{h}}{\cancel{h}} = a
  • Derivate of function f(x) is often marked with the symbol f'(x) (read: "f prim from x").
  • In practice, derivatives are rarely calculated directly from definition. For everyday usage, ready-made derivatives are used.
  • Sometimes it is difficult or impossible to calculate derivative in the analytical form (i.e. given by explicit formula such as -sinx). In this case, the numerical calculations may be alternative. They are based on calculating the derivative value on the selected range using the definition of the difference quotient directly, assuming a finite, but "contractually small" increment value h, e.g. 0.00001. In practice, the value of the increment is selected experimentally for a specific application. In this way you can get approximate values of the derivative, which is often used in natural or engineering sciences.
  • The unique and, due to this fact, very interesting function for mathematicians, is f(x)=exf(x) = e^x. It's derivate is the same as original function.
    (ex)=ex(e^x)' = e^x
  • The derivative can be interpreted as the measure of variability of the function. Such interpretation is particularly useful in natural and engineering sciences, e.g.
    • in physics velocity is a derivative of position over time, so it's the quantity determining how quickly the position of the body changes over time,
    • in electronics, the current intensity is defined as a derivative of the flow of electric charge over time,
    • in chemistry dipole moment is a derivative of the molecule energy over the intensity of the electric field, in other words it tells how much the external electric field will affect the energy of the molecule,
    • etc.

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