Successive Differentiation

Excerpts

• Successive Differentiation

  Begin with a concrete case.

• Successive Differentiation

  This is called the “derived function” of $x$.

• Successive Differentiation

  So the statement $y=f(x)$ merely tells us that $y$ is a function of $x$, it may be $x^2$ or $ax^n$, or $\cos x$ or any other complicated function of $x$.

• Successive Differentiation

  This is to employ the general symbol $f(x)$ for any function of $x$. Here the symbol $f(~)$ is read as “function of,” without saying what particular function is meant. So the statement $y=f(x)$ merely tells us that $y$ is a function of $x$, it may be $x^2$ or $ax^n$, or $\cos x$ or any other complicated function of $x$.

• Successive Differentiation

  The corresponding symbol for the differential coefficient is $f'(x)$, which is simpler to write than $\dfrac{dy}{dx}$. This is called the “derived function” of $x$.

• Successive Differentiation

  Suppose we differentiate over again, we shall get the “second derived function” or second differential coefficient, which is denoted by $f''(x)$; and so on.

• Successive Differentiation

  Similarly, we may write as the result of thrice differentiating, $\dfrac{d^3y}{dx^3} = f'''(x)$.

Equations

• Successive Differentiation

  y = x^5

  The concrete function y = x^5 is differentiated repeatedly as a first example of successive differentiation.

• Successive Differentiation

  y = f(x) = x^n

  A general power function y = f(x) = x^n is introduced to generalise the repeated differentiation.

• Successive Differentiation

  f'(x) &= nx^{n-1}

  The first derivative of x^n is n x^(n-1).

• Successive Differentiation

  f''(x) &= n(n-1)x^{n-2}

  The second derivative of x^n is n(n-1) x^(n-2).

• Successive Differentiation

  f'''(x) &= n(n-1)(n-2)x^{n-3}

  The third derivative of x^n is n(n-1)(n-2) x^(n-3).

• Successive Differentiation

  f''''(x) &= n(n-1)(n-2)(n-3)x^{n-4}

  The fourth derivative of x^n is n(n-1)(n-2)(n-3) x^(n-4).

• Successive Differentiation

  y &= f(x)

  The original function is written y = f(x), so y is a function of x.

• Successive Differentiation

  \frac{dy}{dx} = f'(x)

  The differential coefficient of y with respect to x is denoted f'(x), the derived function.

• Successive Differentiation

  \frac{d\left(\dfrac{dy}{dx}\right)}{dx} &= f''(x)

  Differentiating dy/dx again gives the second derived function f''(x).

• Successive Differentiation

  \dfrac{d^3y}{dx^3} = f'''(x)

  Thrice differentiating y with respect to x is written d^3y/dx^3 and equals f'''(x).

• Successive Differentiation

  y = f(x) = 7x^4 + 3.5x^3 - \frac{1}{2}x^2 + x - 2

  A specific polynomial y = f(x) is chosen as an example for successive differentiation.

• Successive Differentiation

  \frac{dy}{dx} &= f'(x) = 28x^3 + 10.5x^2 - x + 1

  The first derivative of the example polynomial is 28x^3 + 10.5x^2 - x + 1.

• Successive Differentiation

  \frac{d^2y}{dx^2} &= f''(x) = 84x^2 + 21x - 1

  The second derivative of the example polynomial is 84x^2 + 21x - 1.

• Successive Differentiation

  \frac{d^3y}{dx^3} &= f'''(x) = 168x + 21

  The third derivative of the example polynomial is 168x + 21.

• Successive Differentiation

  \frac{d^4y}{dx^4} &= f''''(x) = 168

  The fourth derivative of the example polynomial is the constant 168.

• Successive Differentiation

  \frac{d^5y}{dx^5} &= f'''''(x) = 0

  The fifth derivative of the example polynomial is zero.

• Successive Differentiation

  y = \phi(x) = 3x(x^2 - 4)

  A second example function y = phi(x) = 3x(x^2 - 4) is given for successive differentiation.

• Successive Differentiation

  \phi'(x) &= \frac{dy}{dx} = 3\bigl[x × 2x + (x^2 - 4) × 1\bigr] = 3(3x^2 - 4)

  The first derivative of 3x(x^2 - 4) is 3(3x^2 - 4), obtained by the product rule.

• Successive Differentiation

  \phi''(x) &= \frac{d^2y}{dx^2} = 3 × 6x = 18x

  The second derivative of the second example function is 18x.

• Successive Differentiation

  \phi'''(x) &= \frac{d^3y}{dx^3} = 18

  The third derivative of the second example function is the constant 18.

• Successive Differentiation

  \phi''''(x) &= \frac{d^4y}{dx^4} = 0

  The fourth derivative of the second example function is zero.

Problems