Derivatives – Lessons with Examples

Derivatives are the engine of calculus. They measure how fast things change — and once you master the core rules, you can differentiate almost any function you’ll ever encounter in Calculus 1. This lesson walks through every major rule and technique, with clear explanations and worked examples for each one.

📚 Table of Contents

1. What is a Derivative?
2. The Limit Definition
3. Power Rule
4. Basic Derivative Rules
5. Product Rule
6. Quotient Rule
7. Chain Rule
8. Derivatives of Trig Functions
9. Derivatives of Exponential & Log Functions
10. Implicit Differentiation
11. Higher Order Derivatives
12. Common Mistakes to Avoid
13. Practice Problems

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A derivative measures the instantaneous rate of change of a function. Geometrically, it gives you the slope of the tangent line to the curve at any given point.

Think of it this way: if f(x) describes your position, then f'(x) describes your velocity — how fast your position is changing at any instant.

Notation

All of these mean the same thing — the derivative of f with respect to x:

$$f'(x) \qquad \frac{dy}{dx} \qquad \frac{d}{dx}[f(x)] \qquad \dot{y}$$

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Before using the shortcut rules, it’s important to understand where they come from. The formal definition is:

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

This is the slope formula between two points on the curve, as the distance between them shrinks to zero.

✏️ Example 1 — Using the Definition

Problem: Find f'(x) for \( f(x) = x^2 \) using the limit definition.

Step 1: Write out the formula:

$$f'(x) = \lim_{h \to 0} \frac{(x+h)^2 – x^2}{h}$$

Step 2: Expand:

$$= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 – x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h}$$

Step 3: Factor and cancel:

$$= \lim_{h \to 0} \frac{h(2x + h)}{h} = \lim_{h \to 0} (2x + h) = 2x$$

✅ Answer: \( f'(x) = 2x \)

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The Power Rule is the most used rule in all of Calculus 1. Learn it cold.

$$\frac{d}{dx}[x^n] = nx^{n-1}$$

Bring the exponent down as a coefficient, then subtract 1 from the exponent. That’s it.

✏️ Example 2

Problem: Differentiate \( f(x) = x^5 \)

$$f'(x) = 5x^4$$

✏️ Example 3

Problem: Differentiate \( f(x) = x^{-3} \)

$$f'(x) = -3x^{-4}$$

✏️ Example 4

Problem: Differentiate \( f(x) = \sqrt{x} = x^{1/2} \)

$$f'(x) = \frac{1}{2}x^{-1/2} = \frac{1}{2\sqrt{x}}$$

✏️ Example 5

Problem: Differentiate \( f(x) = \frac{1}{x^4} = x^{-4} \)

$$f'(x) = -4x^{-5} = \frac{-4}{x^5}$$

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These rules let you handle sums, differences, and constants without breaking a sweat.

Rule	Formula	Example
Constant Rule	$$\frac{d}{dx}[c] = 0$$	$$\frac{d}{dx}[7] = 0$$
Constant Multiple Rule	$$\frac{d}{dx}[cf(x)] = cf'(x)$$	$$\frac{d}{dx}[5x^3] = 15x^2$$
Sum Rule	$$\frac{d}{dx}[f+g] = f’+g’$$	$$\frac{d}{dx}[x^2+x^3] = 2x+3x^2$$
Difference Rule	$$\frac{d}{dx}[f-g] = f’-g’$$	$$\frac{d}{dx}[x^4-x] = 4x^3-1$$

✏️ Example 6 — Combining Rules

Problem: Differentiate \( f(x) = 4x^3 – 7x^2 + 2x – 9 \)

$$f'(x) = 12x^2 – 14x + 2$$

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When two functions are multiplied together, you can’t just differentiate each one separately. Use the Product Rule:

$$\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)$$

A helpful way to remember it: “derivative of first times second, plus first times derivative of second.”

✏️ Example 7

Problem: Differentiate \( f(x) = x^2 \cdot \sin(x) \)

Let \( f = x^2 \), \( g = \sin(x) \), so \( f’ = 2x \), \( g’ = \cos(x) \)

$$\frac{d}{dx} = 2x\sin(x) + x^2\cos(x)$$

✏️ Example 8

Problem: Differentiate \( f(x) = (3x + 1)(x^2 – 4) \)

Let \( f = 3x+1 \), \( g = x^2-4 \), so \( f’ = 3 \), \( g’ = 2x \)

$$\frac{d}{dx} = 3(x^2-4) + (3x+1)(2x) = 3x^2 – 12 + 6x^2 + 2x = 9x^2 + 2x – 12$$

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When one function is divided by another, use the Quotient Rule:

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) – f(x)g'(x)}{[g(x)]^2}$$

Memory trick: “Low d-High minus High d-Low, over Low squared.”

✏️ Example 9

Problem: Differentiate \( f(x) = \frac{x^2 + 1}{x – 3} \)

Let \( f = x^2+1 \), \( g = x-3 \), so \( f’ = 2x \), \( g’ = 1 \)

$$f'(x) = \frac{2x(x-3) – (x^2+1)(1)}{(x-3)^2} = \frac{2x^2 – 6x – x^2 – 1}{(x-3)^2} = \frac{x^2 – 6x – 1}{(x-3)^2}$$

✏️ Example 10

Problem: Differentiate \( f(x) = \frac{\sin(x)}{x} \)

$$f'(x) = \frac{\cos(x) \cdot x – \sin(x) \cdot 1}{x^2} = \frac{x\cos(x) – \sin(x)}{x^2}$$

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The Chain Rule handles composite functions — a function inside another function. It’s the most important rule in Calculus 1.

$$\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)$$

In plain English: differentiate the outside function (leaving the inside alone), then multiply by the derivative of the inside.

✏️ Example 11

Problem: Differentiate \( f(x) = (3x + 2)^5 \)

Outside: \( (\cdot)^5 \) → Inside: \( 3x+2 \)

$$f'(x) = 5(3x+2)^4 \cdot 3 = 15(3x+2)^4$$

✏️ Example 12

Problem: Differentiate \( f(x) = \sin(x^2) \)

Outside: \( \sin(\cdot) \) → Inside: \( x^2 \)

$$f'(x) = \cos(x^2) \cdot 2x = 2x\cos(x^2)$$

✏️ Example 13

Problem: Differentiate \( f(x) = e^{3x^2} \)

Outside: \( e^{(\cdot)} \) → Inside: \( 3x^2 \)

$$f'(x) = e^{3x^2} \cdot 6x = 6xe^{3x^2}$$

✏️ Example 14 — Chain Rule + Product Rule

Problem: Differentiate \( f(x) = x^2 \cdot (x^3 + 1)^4 \)

Use Product Rule with Chain Rule on the second factor:

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

$$= 2x(x^3+1)^4 + 12x^4(x^3+1)^3$$

$$= 2x(x^3+1)^3\left[(x^3+1) + 6x^3\right]$$

$$= 2x(x^3+1)^3(7x^3+1)$$

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These six derivatives must be memorized. They appear constantly on exams.

Function	Derivative
$$\sin(x)$$	$$\cos(x)$$
$$\cos(x)$$	$$-\sin(x)$$
$$\tan(x)$$	$$\sec^2(x)$$
$$\csc(x)$$	$$-\csc(x)\cot(x)$$
$$\sec(x)$$	$$\sec(x)\tan(x)$$
$$\cot(x)$$	$$-\csc^2(x)$$

✏️ Example 15

Problem: Differentiate \( f(x) = 3\sin(x) – 2\cos(x) \)

$$f'(x) = 3\cos(x) + 2\sin(x)$$

✏️ Example 16 — Chain Rule with Trig

Problem: Differentiate \( f(x) = \sin(5x^2) \)

$$f'(x) = \cos(5x^2) \cdot 10x = 10x\cos(5x^2)$$

✏️ Example 17

Problem: Differentiate \( f(x) = \tan^3(x) = [\tan(x)]^3 \)

$$f'(x) = 3\tan^2(x) \cdot \sec^2(x)$$

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Function	Derivative
$$e^x$$	$$e^x$$
$$a^x$$	$$a^x \ln(a)$$
$$\ln(x)$$	$$\frac{1}{x}$$
$$\log_a(x)$$	$$\frac{1}{x\ln(a)}$$

✏️ Example 18

Problem: Differentiate \( f(x) = e^{5x} \)

$$f'(x) = e^{5x} \cdot 5 = 5e^{5x}$$

✏️ Example 19

Problem: Differentiate \( f(x) = \ln(x^2 + 1) \)

$$f'(x) = \frac{1}{x^2+1} \cdot 2x = \frac{2x}{x^2+1}$$

✏️ Example 20

Problem: Differentiate \( f(x) = 2^x \)

$$f'(x) = 2^x \ln(2)$$

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When y is not isolated on one side of the equation, you can’t differentiate directly. Instead, differentiate both sides with respect to x, treating y as a function of x and applying the Chain Rule whenever you differentiate a term with y.

Every time you differentiate a y term, multiply by \( \frac{dy}{dx} \).

✏️ Example 21

Problem: Find \( \frac{dy}{dx} \) for \( x^2 + y^2 = 25 \)

Step 1: Differentiate both sides:

$$2x + 2y\frac{dy}{dx} = 0$$

Step 2: Solve for \( \frac{dy}{dx} \):

$$2y\frac{dy}{dx} = -2x \Rightarrow \frac{dy}{dx} = \frac{-x}{y}$$

✏️ Example 22

Problem: Find \( \frac{dy}{dx} \) for \( x^3 + y^3 = 6xy \)

Step 1: Differentiate both sides:

$$3x^2 + 3y^2\frac{dy}{dx} = 6y + 6x\frac{dy}{dx}$$

Step 2: Collect \( \frac{dy}{dx} \) terms:

$$3y^2\frac{dy}{dx} – 6x\frac{dy}{dx} = 6y – 3x^2$$

Step 3: Factor and solve:

$$\frac{dy}{dx}(3y^2 – 6x) = 6y – 3x^2 \Rightarrow \frac{dy}{dx} = \frac{6y – 3x^2}{3y^2 – 6x} = \frac{2y – x^2}{y^2 – 2x}$$

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You can differentiate a function more than once. The second derivative measures how the rate of change is itself changing — in physics, this is acceleration.

Order	Notation	Meaning
1st	\( f'(x) \) or \( \frac{dy}{dx} \)	Rate of change / slope
2nd	\( f”(x) \) or \( \frac{d^2y}{dx^2} \)	Concavity / acceleration
3rd	\( f”'(x) \) or \( \frac{d^3y}{dx^3} \)	Jerk (rate of acceleration)

✏️ Example 23

Problem: Find \( f”(x) \) for \( f(x) = 4x^4 – 3x^3 + 2x \)

$$f'(x) = 16x^3 – 9x^2 + 2$$

$$f”(x) = 48x^2 – 18x$$

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• ❌ Forgetting the Chain Rule: \( \frac{d}{dx}[\sin(x^2)] \neq \cos(x^2) \) — you must multiply by \( 2x \)
• ❌ Using Product Rule on a constant: \( \frac{d}{dx}[5x^3] = 15x^2 \), not a product rule situation
• ❌ Quotient Rule sign error: It’s always top derivative times bottom minus top times bottom derivative — order matters!
• ❌ Forgetting \( \frac{dy}{dx} \) in implicit differentiation: Every y term needs it
• ❌ Derivative of \( e^x \) with chain rule: \( \frac{d}{dx}[e^{3x}] = 3e^{3x} \), not just \( e^{3x} \)
• ❌ Power Rule on exponentials: \( \frac{d}{dx}[e^x] \neq xe^{x-1} \) — the Power Rule only applies to \( x^n \)

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Try these before checking the answers!

1.

$$\frac{d}{dx}[6x^4 – 3x^2 + 7]$$

👁️ Show Solution

$$24x^3 – 6x$$

2.

$$\frac{d}{dx}[x^3 \cdot e^x]$$

👁️ Show Solution

$$3x^2 e^x + x^3 e^x = x^2 e^x(3 + x)$$

3.

$$\frac{d}{dx}\left[\frac{x^2 + 3}{2x – 1}\right]$$

👁️ Show Solution

$$\frac{2x(2x-1) – (x^2+3)(2)}{(2x-1)^2} = \frac{4x^2 – 2x – 2x^2 – 6}{(2x-1)^2} = \frac{2x^2 – 2x – 6}{(2x-1)^2}$$

4.

$$\frac{d}{dx}[(x^2 + 5)^6]$$

👁️ Show Solution

$$6(x^2+5)^5 \cdot 2x = 12x(x^2+5)^5$$

5.

$$\frac{d}{dx}[\ln(3x^2 + 1)]$$

👁️ Show Solution

$$\frac{6x}{3x^2+1}$$

6. Find \( \frac{dy}{dx} \) for \( x^2 + 3y^2 = 12 \)

👁️ Show Solution

$$2x + 6y\frac{dy}{dx} = 0 \Rightarrow \frac{dy}{dx} = \frac{-x}{3y}$$

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