Skip to main content Skip to page footer
TechMechAcademy logo TechMechAcademy logo

Derivatives of Basic Functions

In technical mechanics, the analysis of functional relationships plays a crucial role. Mathematical functions are often used to describe movements or forces, for example. The derivatives of these functions allow us to calculate important properties such as velocities, accelerations, or forces.

Here, derivatives of basic functions are listed, which frequently serve as a starting point for further calculations.

Derivatives of Basic Functions

Table 1: Derivatives of basic functions
Function \(y\) Derivative \(\dot{y}\)
Constant Function \(C =\) constant \(0\)
Power Functions \(t^n\) \(n \cdot t^{n-1}\)
\(\sqrt{t} = t^{\frac{1}{2}}\) \(\dfrac{1}{2} \cdot t^{\frac{1}{2}-1} = \dfrac{1}{2} \cdot t^{-\frac{1}{2}} = \dfrac{1}{2\sqrt{t}}\)
Trigonometric Functions \(\sin(t)\) \(\cos(t)\)
\(\cos(t)\) \(-\sin(t)\)
\(\tan(t)\) \(\dfrac{1}{\cos^2(t)}= 1 + \tan^2(t)\)
\(\cot(t)\) \(-\dfrac{1}{\sin^2(t)}= -1 - \cot^2(t)\)
Inverse Functions \(\sin^{-1}(t)\) \(\dfrac{1}{\sqrt{1-t^2}}\)
\(\cos^{-1}(t)\) \(-\dfrac{1}{\sqrt{1-t^2}}\)
\(\tan^{-1}(t)\) \(\dfrac{1}{1+t^2}\)
\(\cot^{-1}(t)\) \(-\dfrac{1}{1+t^2}\)
Exponential Functions \(e^t\) \(e^t\)
\(a^t\) \(\ln(a) \cdot a^t\)
Logarithmic Functions \(\ln(t)\) \(\dfrac{1}{t}\)
\(\log_a(t)\) \(\dfrac{1}{\ln(a) \cdot t}\)
Hyperbolic Functions \(\sinh(t)\) \(\cosh(t)\)
\(\cosh(t)\) \(\sinh(t)\)
\(\tanh(t)\) \(\dfrac{1}{\cosh^2(t)}= 1 - \tanh^2(t)\)
\(\coth(t)\) \(-\dfrac{1}{\sinh^2(t)}= 1 - \coth^2(t)\)
Areafunktionen \(\sinh^{-1}(t)\) \(\dfrac{1}{\sqrt{t^2+1}}\)
\(\cosh^{-1}(t)\) \(\dfrac{1}{\sqrt{t^2-1}}\)
\(\tanh^{-1}(t)\) \(\dfrac{1}{1-t^2}\)
\(\coth^{-1}(t)\) \(\dfrac{1}{1-t^2}\)