Formulas and Tables

Calculus

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 \(x.\)

Differential of a function
  1. The differential of the independent variable is equal to its increment: \(dx = \Delta x\)
  2. The differential of a function is equal to the derivative of the function times the differential of the independent variable:
    \(dy = df\left( x \right) =\) \( f’\left( x \right)dx\)
  3. Derivative as the quotient of two differentials
    \(f’\left( x \right) = \large\frac{{dy}}{{dx}}\normalsize\)
  4. The differential of a constant is zero: \(dC = 0\)
  5. differential of the sum of two functions is equal to the sum of their differentials:
    \(d\left( {u + v} \right) =\) \( du + dv\)
  6. differential of the difference of two functions is equal to the difference of their differentials:
    \(d\left( {u – v} \right) =\) \( du – dv\)
  7. constant factor can be taken out of the differential:
    \(d\left( {Cu} \right) = Cdu\)
  8. Differential of the product of two functions
    \(d\left( {uv} \right) = vdu + udv\)
  9. Differential of the quotient of two functions
    \(d\left( {{\large\frac{u}{v}}\normalsize} \right) = {\large\frac{{vdu – udv}}{{{v^2}}}\normalsize}\)