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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} \frac{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} \frac{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} \frac{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} \frac{a \cdot (x + h) + b - (ax + b)}{h} = \lim\limits_{h \to 0} \frac{\cancel{ax} + ah + \cancel{b} - \cancel{ax} - \cancel{b}}{h} = \lim\limits_{h \to 0} \frac{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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