# Derivatives as local rate of change

Physics I · Calculus of a Single Variable · https://tryals.app/learn/physics-i/derivatives-as-local-rate-of-change

## Tangent Lines and Meaning

The **derivative** of $f$ at $x$ gives the instantaneous rate of change and is defined as the limit of the difference quotient:

$$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

**Geometric meaning**: $f'(a)$ is the **slope of the tangent line** at $(a, f(a))$. The tangent line equation is:

$$y - f(a) = f'(a)(x - a)$$

This formula acts as the best **linear approximation** of $f$ near $a$.

**Physical meaning**: If $s(t)$ is position, its first derivative $s'(t)$ is **velocity** and its second derivative $s''(t)$ is **acceleration**.

**Sign behavior**: Where $f'(x) > 0$ the function is **increasing**; where $f'(x) < 0$ it is **decreasing**; $f'(x) = 0$ marks critical points.

**Common pitfall**: The derivative at a point is a **number** (a slope), while the derivative of a function is a new **function**. Confusing $f'(3)$ with $f'(x)$ causes common errors.

## Rules and Differentiability

Quickly compute common derivatives using these foundational rules:

| $f(x)$ | $f'(x)$ |
| --- | --- |
| $x^n$ | $nx^{n-1}$ |
| $\sin x$ | $\cos x$ |
| $e^{x}$ | $e^{x}$ |
| $\ln x$ | $1/x$ |

**Differentiability vs continuity** compares whether smooth rates exist versus unbroken paths:

| Property | Rule | Example |
| --- | --- | --- |
| Differentiable | $\Rightarrow$ Continuous | Polynomials |
| Continuous | $\not\Rightarrow$ Differentiable | $|x|$ at $x=0$ |

A function must be continuous to be differentiable, but a sharp corner breaks differentiability while remaining continuous.

## Practice questions

3 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Your car’s **speedometer** reads 60 km/h at this instant. Which calculus statement matches that reading?

A. Your acceleration is currently 60 km/h
B. The average of distance over the whole trip is 60 km/h
C. The derivative of position with respect to time
D. You will travel 60 km in the next hour

**Answer:** C. The derivative of position with respect to time

**Why:** The speedometer is a physical derivative-meter: it shows the limit of $\Delta x/\Delta t$ over vanishing intervals. Averages need a whole interval; the derivative compresses that idea into a single instant.

Page: https://tryals.app/practice/physics-i/derivatives-as-local-rate-of-change/your-cars-speedometer-reads-60-km-h-at-this-instant-which-calculus

### 2. Position, velocity and acceleration are successive derivatives. Why does a car with zero velocity at the top of a hill not also have zero acceleration?

A. Because acceleration is always nonzero for a moving object
B. Because gravity acts on the car regardless of what the mathematics says
C. Because zero velocity says nothing about how it is CHANGING
D. Because zero velocity means the car has stopped, so acceleration is undefined

**Answer:** C. Because zero velocity says nothing about how it is CHANGING

**Why:** A value and its rate of change are independent at an instant. This is the same reason a ball thrown upward has $a = -g$ at the top of its arc even though it is momentarily at rest.

Page: https://tryals.app/practice/physics-i/derivatives-as-local-rate-of-change/position-velocity-and-acceleration-are-successive-derivatives-why

### 3. Which statements are correct when f'(x) > 0 on an interval?

A. f has a local minimum somewhere on that interval
B. f is increasing on that interval
C. The tangent line has positive slope
D. f is concave up on that interval

**Answer:** B. f is increasing on that interval; C. The tangent line has positive slope

**Why:** f' > 0 ⟹ increasing and positive-sloped tangents. Concavity depends on f'', not f'.

Page: https://tryals.app/practice/physics-i/derivatives-as-local-rate-of-change/which-statements-are-correct-when-f-x-0-on-an-interval