# The Derivative

Mathematics I · Differential Calculus · https://tryals.app/learn/mathematics-i/the-derivative

## What is a Derivative?

The **derivative** measures how fast a function changes at an instant. It is the limit of **average rates of change** as intervals shrink:

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

The fraction is the slope of the **secant line**. As $h \to 0$, it becomes the **tangent line**.

| Term | Meaning |
| --- | --- |
| **Derivative** | Instantaneous rate of change & slope of tangent |
| **Secant Line** | Line through two points on the graph |
| **Tangent Line** | Limit of secant as points merge |

Notation includes $f'(x)$, $\frac{df}{dx}$, $\frac{dy}{dx}$, and $Df$. A function is **differentiable** if this limit exists.

> **Pitfall:** Do not confuse the **derivative** (slope) with the **function value** (height). $f'(x) = 0$ means the graph is **flat**, not zero.

## Behavior & Differentiability

The sign of the **derivative** tells you the function's local behavior:

| Derivative Sign | Function Behavior | Tangent State |
| --- | --- | --- |
| $f'(x) > 0$ | **Increasing** | Sloping upward |
| $f'(x) < 0$ | **Decreasing** | Sloping downward |
| $f'(x) = 0$ | **Candidate peak/valley** | **Horizontal** |

Differentiability is **stronger than continuity**. Every differentiable function is continuous, but not conversely.

A **sharp corner** (like $|x|$ at 0) is continuous yet **not differentiable** because left and right slopes disagree. Vertical tangents and discontinuities also block differentiability.

## Practice questions

4 of this lesson's 10 practice questions, with answers. The full set is in the app.

### 1. Select every TRUE statement about the derivative.

A. A sharp corner blocks differentiability
B. f'(x) = 0 means that f(x) = 0
C. Every continuous function is differentiable
D. Every differentiable function is continuous
E. How steeply the curve rises at x is exactly what f'(x) reports

**Answer:** A. A sharp corner blocks differentiability; D. Every differentiable function is continuous; E. How steeply the curve rises at x is exactly what f'(x) reports

**Why:** Differentiability implies continuity but not conversely, the absolute value is the standard witness, and its corner is exactly the obstruction. And f'(x) = 0 says the graph is momentarily FLAT at x, saying nothing whatever about its height.

Page: https://tryals.app/practice/mathematics-i/the-derivative/select-every-true-statement-about-the-derivative

### 2. The derivative at a point is the slope of the secant line through two nearby points on the curve.

**Answer:** False

**Why:** False. That is the difference *quotient*, an approximation; the derivative is its **limit** as the two points merge, which turns the secant into the tangent. The distinction matters because the limit can fail to exist, at a corner the secants approach different slopes from each side and there is no derivative.

Page: https://tryals.app/practice/mathematics-i/the-derivative/the-derivative-at-a-point-is-the-slope-of-the-secant-line-through-two

### 3. The derivative is defined as a limit of 0/0, the difference quotient is undefined at h = 0 for every function. Why is that not a fatal problem?

A. Because the difference quotient is defined at h = 0 for differentiable functions
B. Because a limit deliberately excludes the point itself
C. Because h is never actually allowed to reach zero in practice
D. Because the numerator also goes to zero, and the two cancel exactly

**Answer:** B. Because a limit deliberately excludes the point itself

**Why:** This is what limits were for. The quotient is 0/0 at h = 0 for every function ever differentiated, so a definition requiring the point would define nothing, the limit describes the approach instead. Newton and Leibniz used the idea for a century before Cauchy and Weierstrass made it rigorous.

Page: https://tryals.app/practice/mathematics-i/the-derivative/the-derivative-is-defined-as-a-limit-of-0-0-the-difference-quotient

### 4. Order the steps of computing f'(x) for $f(x) = x^{2}$ from the limit definition.

**Answer:**

1. Write the difference quotient ((x + h) squared - x squared)/h
2. Expand the numerator to 2xh + h squared
3. Cancel one factor of h to leave 2x + h
4. Let h approach 0 to obtain 2x

**Why:** Substituting h = 0 before cancelling gives 0/0 and no information; cancelling first is legal because h is never actually 0 in the limit process. Every derivative from first principles follows this same pattern: expand, cancel the h, then take the limit.

Page: https://tryals.app/practice/mathematics-i/the-derivative/order-the-steps-of-computing-f-x-for-f-x-x-from-the-limit