Formulas

Calculus

Calculus Formulas Logo

Basic Applications of the Derivative

  • Functions: \(f\), \(g\), \(y\)
    Argument (independent variable): \(x\)
    Point coordinates: \({x_0},\) \({y_0},\) \({x_1},\) \({x_2},\) \({x_3}\)
    Real numbers: \(a\), \(b\), \(c\)
    Position of an object: \(s\)
    Velocity: \(v\)
    Acceleration: \(w\)
    Time: \(t\)
    1. Velocity and acceleration
      Suppose that the function \(s\left( t \right)\) describes the position of an object in a coordinate system at time \(t\). Then the first derivative of the function \(s\left( t \right)\) is the instantaneous velocity of the object:
      \(v = s^{\,\prime} = f'{\left( t \right)}\)
      The second derivative of the function \(s\left( t \right)\) is the instantaneous acceleration of the object:
      \(w = v^{\,\prime} = s^{\,\prime\prime} =\) \(f^{\prime\prime}{\left( t \right)}\)
    2. Equation of a tangent
      \(y – {y_0} =\) \(f^\prime {\left( {x_0} \right)} {\left( x – {x_0}\right)},\)
      where \(\left( {x_0},{y_0} \right)\) are the coordinates of the point of tangency, \(f^\prime {\left( {x_0} \right)}\) is the value of the derivative of the function \(f\left( x \right)\) at the point of tangency.
    3. Equation of a tangent and a normal
    4. Equation of a normal
      \(y – {y_0} =\) \( – {\large\frac{1}{{f’\left( {{x_0}} \right)}}\normalsize} \left( {x – {x_0}} \right),\)
      where \(\left( {x_0},{y_0} \right)\) are the coordinates of the point of intersection, \(f^\prime {\left( {x_0} \right)}\) is the value of the derivative of the function \(f\left( x \right)\) at this point.
    5. Increasing and decreasing functions
      If \(f’\left( {{x_0}} \right) \gt 0\), then the function \(f\left( x \right)\) is increasing at \({x_0}\). In the figure below, the function increases at \(x \lt {x_1}\) and \(x \gt {x_2}\).
      If \(f’\left( {{x_0}} \right) \lt 0\), then the function \(f\left( x \right)\) is decreasing at \({x_0}\) (the interval \(\left.{{x_1} \lt x \lt {x_2}}\right).\)
      If \(f’\left( {{x_0}} \right) = 0\) or the derivative does not exist, then the test fails.
    6. Local extrema and inflection point
    7. Local extrema of a function
      A function \(f\left( x \right)\) has a local maximum at a point \({x_1}\) if and only if there exists some interval containing \({x_1}\) such that \(f\left( {{x_1}} \right) \ge f\left( x \right)\) for all \(x\) in this interval.
      Similarly, a function \(f\left( x \right)\) has a local minimum at a point \({x_2}\) if and only if there exists some interval containing \({x_2}\) such that \(f\left( {{x_2}} \right) \le f\left( x \right)\) for all \(x\) in this interval.
    8. Critical points
      A point \({x_0}\) is called a critical point of the function \(f\left( x \right)\) if and only if either \(f’\left( {x_0} \right)\) is zero or does not exist.
    9. First derivative test for local extrema
      If \(f\left( x \right)\) is increasing (\(f’\left( x \right) \gt 0\)) for all \(x\) in some interval \(\left( {a,{x_1}} \right]\) and decreasing \(\left({f’\left( x \right) \lt 0}\right)\) for all \(x\) in some interval \(\left[ {{x_1},b} \right)\), then the function \(f\left( x \right)\) has a local maximum at the point \({x_1}\).
      Similarly, if \(f\left( x \right)\) is decreasing \(\left({f’\left( x \right) \lt 0}\right)\) for all \(x\) in some interval \(\left( {a,{x_2}} \right]\) and increasing \(\left({f’\left( x \right) \gt 0}\right)\) for all \(x\) in some interval \(\left[ {{x_2},b} \right)\), then the function \(f\left( x \right)\) has a local minimum at the point \({x_2}\).
    10. Second derivative test for local extrema
      If \(f’\left( {{x_1}} \right) = 0\) and \(f^{\prime\prime}\left( {{x_1}} \right) \lt 0,\) then the function \(f\left( x \right)\) attains a local maximum at the point \({x_1}.\)
      If \(f’\left( {{x_2}} \right) = 0\) and \(f^{\prime\prime}\left( {{x_2}} \right) \gt 0,\) then the function \(f\left( x \right)\) attains a local minimum at the point \({x_2}.\)
    11. Concavity of a function. First derivative test for concavity
      A function \(f\left( x \right)\) is concave upward at a point \({x_0}\) if and only if the derivative \(f’\left( x \right)\) is increasing at this point (the interval \(x \lt {x_3}\) in the figure above).
      Similarly, a function \(f\left( x \right)\) is concave downward at a point \({x_0}\) if and only if the derivative \(f’\left( x \right)\) is decreasing at this point (the interval \(x \gt {x_3}\)).
    12. Second derivative test for concavity
      If \(f^{\prime\prime}\left( {{x_0}} \right) \gt 0\), then the function \(f\left( x \right)\) is concave upward at \({x_0}\).
      If \(f^{\prime\prime}\left( {{x_0}} \right) \lt 0\), then the function \(f\left( x \right)\) is concave downward at \({x_0}\).
      If \(f^{\prime\prime}\left( {{x_0}} \right) = 0\) or does not exist at \({x_0},\) then the test fails at this point.
    13. Inflection point
      If the first derivative\(f’\left( {x_3} \right)\) exists at a point \({x_3}\) and the second derivative \(f^{\prime\prime}\left( {x_3} \right)\) changes sign at \(x = {x_3}\), then the point \(\left( {{x_3},f\left( {{x_3}} \right)} \right)\) is called an inflection point of the graph of the function \(f\left( x \right)\).
      If the second derivative \(f^{\prime\prime}\left( {x_3} \right)\) exists at the inflection point, then \(f^{\prime\prime}\left( {x_3} \right) = 0\).
    14. L’Hopital’s rule
      \(\lim\limits_{x \to c} {\large\frac{{f\left( x \right)}}{{g\left( x \right)}}\normalsize} = \lim\limits_{x \to c} {\large\frac{{f’\left( x \right)}}{{g’\left( x \right)}}\normalsize},\) \(\text { if }\) \(\lim\limits_{x \to c} f\left( x \right) =\) \(\lim\limits_{x \to c} g\left( x \right) = \left[ {\begin{array}{*{20}{c}} 0\\ \infty \end{array}} \right..\)