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Definition and Properties of the Derivative

  • Functions: \(f\), \(g\), \(y\), \(u\), \(v\)
    Argument (independent variable): \(x\)
    Real constant: \(k\)
    Angle: \(\alpha\)
    1. The derivative of a function \(y = f\left( x \right)\) measures the rate of change of \(y\) with respect to \(x\). Suppose that at some point \(x \in \mathbb{R}\), the argument of a continuous real function \(y = f\left( x \right)\) has an increment \(\Delta x\). Then the increment of the function is equal to
      \(\Delta y = f\left( {x + \Delta x} \right) \,-\) \(f\left( x \right).\)
      The derivative of the function \(y = f\left( x \right)\) at the point \(x\) is defined as the limit of the ratio \(\Delta y/\Delta x\) as \(\Delta x \to 0:\)
      \(y’ = f’\left( x \right) = {\large\frac{{dy}}{{dx}}\normalsize} =\) \({\large\frac{{df\left( x \right)}}{{dx}}\normalsize} =\) \(\lim\limits_{x \to 0} {\large\frac{{\Delta y}}{{\Delta x}}\normalsize} =\) \(\lim\limits_{x \to 0} {\large\frac{{f\left( {x + \Delta x} \right) – f\left( x \right)}}{{\Delta x}}\normalsize}.\)
    2. Definition of the derivative of a function
    3. From a geometrical point of view, the derivative of a function \(y = f\left( x \right)\) at a point \(x\) is equal to the slope of the tangent line to the curve \(f\left( x \right)\) drawn through this point:
      \({\large\frac{{dy}}{{dx}}\normalsize} = \tan \alpha \)
    4. The derivative of the sum of two functions is equal to the sum of their derivatives:
      \({\large\frac{{d\left( {u + v} \right)}}{{dx}}\normalsize} = {\large\frac{{du}}{{dx}}\normalsize} + {\large\frac{{dv}}{{dx}}\normalsize}\)
    5. The derivative of the difference of two functions is equal to the difference of their derivatives:
      \({\large\frac{{d\left( {u – v} \right)}}{{dx}}\normalsize} = {\large\frac{{du}}{{dx}}\normalsize} – {\large\frac{{dv}}{{dx}}\normalsize}\)
    6. A constant factor can be taken out of a derivative:
      \({\large\frac{{d\left( {ku} \right)}}{{dx}}\normalsize} = k{\large\frac{{du}}{{dx}}\normalsize}\)
    7. Derivative of the product of two functions
      \({\large\frac{{d\left( {u \cdot v} \right)}}{{dx}}\normalsize} =\) \( {\large\frac{{du}}{{dx}}\normalsize} \cdot v + u \cdot {\large\frac{{dv}}{{dx}}\normalsize}\)
    8. Derivative of the quotient of two functions
      \({\large\frac{d}{{dx}}\normalsize}\left( {{\large\frac{u}{v}}\normalsize} \right) =\) \({\large\frac{{\frac{{du}}{{dx}} \cdot v – u \cdot \frac{{dv}}{{dx}}}}{{{v^2}}}\normalsize}\)
    9. Derivative of a composite function (chain rule)
      \(y = f\left( {g\left( x \right)} \right),\) \(u = g\left( x \right),\) \({\large\frac{{dy}}{{dx}}\normalsize} = {\large\frac{{dy}}{{du}}\normalsize} \cdot {\large\frac{{du}}{{dx}}\normalsize}\)
    10. Derivative of an inverse function
      \({\large\frac{{dy}}{{dx}}\normalsize} = {\large\frac{1}{{dx/dy}}\normalsize},\)
      where \(x\left( y \right)\) is the inverse function for \(y\left( x \right).\)
    11. \({\large\frac{d}{{dx}}\normalsize} \left( {{\large\frac{1}{y}}\normalsize} \right) =\) \( – {\large\frac{{dy/dx}}{{{y^2}}}\normalsize}\)
    12. Logarithmic derivative
      \(y = f\left( x \right),\) \(\ln y = \ln f\left( x \right),\) \({\large\frac{{dy}}{{dx}}\normalsize} =\) \(f\left( x \right) \cdot {\large\frac{d}{{dx}}\normalsize} \left[ {\ln f\left( x \right)} \right]\)