5.3 Contents

1. Derivative Definition & Function

2. Notation

3. Tangent Line 🔧

4. Differentiability 🔧

5. First Derivative Use

6. Second Derivative Use

5.3.1 Derivative Definition & Function

A function's instantaneous rate of change along any given point. The derivative function is essentially the slope formula with one point. $$\frac{d}{dx}f(x)=\lim\limits_{x\to a}\frac{f(x)-f(a)}{x-a}$$ $$=\lim\limits_{\Delta x→0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

5.3.2 Notation

Notation	First	Second
Leibniz	$$\frac{dy}{dx} \medspace\text{or}\medspace \frac{d}{dx}f{x}$$	$$\frac{d^2y}{dx^2} \medspace\text{or}\medspace \frac{d^2}{dx^2}f{x}$$
Langrange	$$f'(x)$$	$$f''(x)$$
Newton	$$\dot{x}$$	$$\ddot{x}$$
Euler	$$Df$$	$$D^2f$$

5.3.3 Tangent Line 🔧

The point-slope equation for a line parallel to and touching a function at a specific point $$y-f(a)=f'(a)(x-a)$$

5.3.4 Differentiability 🔧

A function $f(x)$ is differentiable at a point $a$ except in the following conditions

• $f(x)$ is not continuous at $a$
• $f(x)$ has a corner (change in function) at $a$
• $f(x)$ has a vertical tangent at $a$

5.3.5 First Derivative Use

• If $f'(a)=0$, then $f(a)$ is a critical point (local maxima or minima)
• If $f'(a) \gt 0$, then $f(a)$ is increasing
• If $f'(a) \lt 0$, then $f(a)$ is decreasing
• If $f'(x)$ doesn’t change signs, then $f(x)$ is monotonic (only increasing or decreasing)

5.3.6 Second Derivative Use

• If $f''(a)=0$, then $f(a)$ is an inflection point (change in concavity)
• If $f''(a)\lt 0$, then $f(a)$ is concave (downwards)
• If $f''(a)\gt 0$, then $f(a)$ is convex (concave upwards)