10D

The notation dy/dx for the derivative

Small changes Δx and Δy, Leibniz notation dy/dx, operator notation d/dx, and second and higher derivatives.

Small changes and dy/dx

When $x$ changes by a small amount $\Delta x$, $y$ changes by $\Delta y$. The ratio $\dfrac{\Delta y}{\Delta x}$ is the gradient of a secant. As $\Delta x \to 0$ it becomes the gradient of the tangent:

Leibniz notation

$$\frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

$\dfrac{dy}{dx}$ and $f'(x)$ mean the same thing. The operator $\dfrac{d}{dx}$ means "differentiate with respect to $x$", as in $\dfrac{d}{dx}(x^3) = 3x^2$. Other letters work too: if $s = 3t^2$, then $\dfrac{ds}{dt} = 6t$.

Second and higher derivatives

Differentiating again gives the second derivative, written $f''(x)$ or $\dfrac{d^2y}{dx^2}$. It measures how fast the gradient itself is changing.

Example 1●

Find $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ for $y = x^4 - 3x^2$.

$\dfrac{dy}{dx} = 4x^3 - 6x$

$\dfrac{d^2y}{dx^2} = 12x^2 - 6$

Example 2●●

Find the third derivative of $x^5$.

$5x^4 \to 20x^3 \to 60x^2$

Flashcards

$dfrac{dy}{dx}$

The derivative of $y$ with respect to $x$.

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

The second derivative, $f''(x)$.

$dfrac{Delta y}{Delta x}$

Gradient of a secant.

Practice

$\dfrac{dy}{dx}$ means:
For $y = x^4 - 3x^2$, find $\dfrac{d^2y}{dx^2}$ at $x = 1$.
The third derivative of $x^5$ is:
Find $\dfrac{d}{dx}(5x^2 - 3x)$ at $x = 2$.
$\dfrac{\Delta y}{\Delta x}$ is the gradient of:
If $f(x) = 2x^3$, find $f''(-1)$.
If $s = 3t^2 - t^3$, find $\dfrac{ds}{dt}$ at $t = 1$.
For which function is $f''(x) = 0$ for every $x$?
For $y = x^3 - 6x^2$, find $x$ where $\dfrac{d^2y}{dx^2} = 0$.