Second and higher derivatives

The second, third and higher derivatives of a function, their notations, and finding successive derivatives of polynomials, negative powers and composite functions.

Differentiating again

The derivative $f'(x)$ is itself a function, so it can be differentiated. The result is the second derivative, which measures how fast the gradient is changing.

	Function notation	Leibniz notation
first	$f'(x)$ or $y'$	$\dfrac{dy}{dx}$
second	$f''(x)$ or $y''$	$\dfrac{d^2y}{dx^2}$
third	$f'''(x)$ or $y'''$	$\dfrac{d^3y}{dx^3}$

Example 1: a polynomial●

Find all the derivatives of $y = x^5 - 2x^3$.

$y' = 5x^4 - 6x^2$, $\;y'' = 20x^3 - 12x$, $\;y''' = 60x^2 - 12$.

$y'''' = 120x$, then $120$, then $0$ from then on.

Example 2: negative powers●●

Find the first three derivatives of $y = \dfrac{1}{x}$.

$y = x^{-1}$, so $y' = -x^{-2}$, $\;y'' = 2x^{-3}$, $\;y''' = -6x^{-4} = -\dfrac{6}{x^4}$.

Example 3: the chain rule twice●●

Find $f''(x)$ for $f(x) = (2x + 1)^4$.

$f'(x) = 4 \times 2 \times (2x + 1)^3 = 8(2x + 1)^3$

$f''(x) = 24 \times 2 \times (2x + 1)^2 = 48(2x + 1)^2$

Flashcards

$dfrac{d^2y}{dx^2}$

The derivative of $\dfrac{dy}{dx}$.

$dfrac{d^2}{dx^2},x^n$

$n(n - 1)x^{n-2}$

Degree $n$ polynomial

Its $(n + 1)$th derivative is $0$.

Practice

For $f(x) = x^5 - 2x^3$, find $f''(1)$.
For $y = \dfrac1x$, find $y'''$ at $x = 1$.
For $f(x) = (2x + 1)^4$, find $f''(0)$.
Find the fifth derivative of $x^4$.
$\dfrac{d^2}{dx^2}\,x^n = $
For $f(x) = x^3$, find $f''(2)$.
For $y = \sqrt{x}$, find $y''$ at $x = 4$.
For $y = x^2$, $\left(\dfrac{dy}{dx}\right)^2 = $
For $y = x^4$, find $\dfrac{d^4y}{dx^4}$.