Formulas

Calculus

Calculus Formulas Logo

Properties of Differentials

  • Functions: \(f\), \(u\), \(v\)
    Argument (independent variable): \(x\)
    Derivatives: \(y’\left( x \right),\) \(f’\left( x \right)\)
    Constant: \(C\)
    Real numbers: \(A\), \(\alpha\)
    Small change in \(y:\) \(\Delta y\)
    Small change in \(x:\) \(\Delta x\)
    Differential of the function \(y:\) \(dy\)
    Differential of the independent variable \(x:\) \(dx\)
    1. Consider a function \(y = f\left( x \right)\) and suppose that the independent variable gets an increment \(dx\) at some point \(x\). This increment is called the differential of the independent variable. The function \(y = f\left( x \right)\) has a differential at the point \(x\) if its increment can be represented as the sum of two terms:
      \(\Delta y = f\left( {x + \Delta x} \right) – f\left( x \right) =\) \( A\Delta x + \alpha ,\)
      where the coefficient \(A\) is independent of \(\Delta x\) and the value of \(\alpha\) has a higher order of smallness with respect to the increment \(\Delta x\), i.e. \(\alpha /\Delta x \to 0\) as \(\Delta x \to 0\).

      In the formula above, the principal linear part of the increment is called the differential of the function \(f\left( x \right)\) at the point \(x\) and denoted by \(dy = A\Delta x\). In this expression the coefficient \(A\) is equal to the value of the derivative \(f’\left( x \right)\) at the point \(x.\)
    2. Differential of a function
    3. The differential of the independent variable is equal to its increment: \(dx = \Delta x\)
    4. The differential of a function is equal to the derivative of the function times the differential of the independent variable:
      \(dy = df\left( x \right) =\) \( f’\left( x \right)dx\)
    5. Derivative as the quotient of two differentials
      \(f’\left( x \right) = \large\frac{{dy}}{{dx}}\normalsize\)
    6. The differential of a constant is zero: \(dC = 0\)
    7. differential of the sum of two functions is equal to the sum of their differentials:
      \(d\left( {u + v} \right) =\) \( du + dv\)
    8. differential of the difference of two functions is equal to the difference of their differentials:
      \(d\left( {u – v} \right) =\) \( du – dv\)
    9. constant factor can be taken out of the differential:
      \(d\left( {Cu} \right) = Cdu\)
    10. Differential of the product of two functions
      \(d\left( {uv} \right) = vdu + udv\)
    11. Differential of the quotient of two functions
      \(d\left( {{\large\frac{u}{v}}\normalsize} \right) = {\large\frac{{vdu – udv}}{{{v^2}}}\normalsize}\)