Introduction to Calculus — Limits, Derivatives & Integrals (Beginner Friendly)

[Leejohnston]

Introduction to Calculus — Limits, Derivatives & Integrals (Beginner Friendly)

Calculus is one of the most powerful areas of mathematics. 
It lets us measure change, movement, growth, decay, curves, areas, and anything that varies smoothly.

This thread introduces the essentials in a simple way, perfect for GCSE extension, A-Level preparation, or beginners in physics and engineering.

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1. What Is Calculus?

Calculus has two main parts:

1. Differentiation — finding the rate of change 
• slopes of graphs 
• gradients 
• how fast something is moving

2. Integration — finding the total 
• area under curves 
• accumulated values 
• distance travelled from speed 

Together, they allow us to understand motion, forces, energy, economics, biology, and more.

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2. Limits — The Foundation of Calculus

A limit describes what a function approaches as x gets close to some value.

Example:

Code:

lim (x→2) (3x + 1) = 7

Because as x gets closer to 2, 3x + 1 gets closer to 7.

Limits allow us to define gradients precisely.

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3. The Derivative — Rate of Change

If y = f(x), the derivative f’(x) tells us:

• how fast y changes 
• the gradient at any point 
• whether the function is increasing or decreasing 

Basic rules:

Power rule: 
d/dx (xⁿ) = n·xⁿ⁻¹

Examples:

Code:

d/dx (x³) = 3x²
d/dx (5x²) = 10x
d/dx (7x) = 7

Derivative of constants: 
d/dx (k) = 0

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4. Common Derivatives

Code:

d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (e^x) = e^x
d/dx (ln x) = 1/x

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5. Using Derivatives — Tangents & Turning Points

Gradient of the tangent: 
Use f’(x) at that point.

Example:

y = x² 
y’ = 2x 
At x = 3 → gradient = 6

Turning points: 
• solve f’(x) = 0 
• plug back into the original function to find coordinates 

Example:

f(x) = x² - 4x 
f’(x) = 2x - 4 
Set to zero: 2x - 4 = 0 → x = 2

Turning point at x = 2.

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6. Integration — Area Under Curves

Integration reverses differentiation.

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C

Examples:

Code:

∫ x dx = x²/2 + C
∫ 3x² dx = x³ + C
∫ 6 dx = 6x + C

Definite integrals find exact area:

Code:

∫(1→4) x dx
= [x²/2] (1→4)
= (16/2) - (1/2)
= 15/2

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7. Applications of Calculus

Calculus appears everywhere in science:

Physics: 
• velocity from displacement 
• acceleration from velocity 
• force from energy 
• motion equations 

Biology: 
• population growth 
• rates of reaction 

Chemistry: 
• decay laws 
• diffusion models 

Economics: 
• marginal cost 
• optimisation 

Engineering: 
• stresses 
• electrical circuits 
• fluid dynamics 

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8. Common Mistakes

❌ Forgetting to add +C when integrating 
✔ Always include the constant for indefinite integrals

❌ Mixing up power rule directions 
✔ Differentiation lowers the power, integration raises it

❌ Thinking gradient = f(x) 
✔ Gradient = f’(x)

❌ Not substituting correctly in definite integrals 
✔ Always write [F(b) – F(a)]

❌ Treating turning points without checking type 
✔ minimum vs maximum → use second derivative if needed

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9. Practice Questions

Differentiation 
1. d/dx (4x³) 
2. d/dx (7x² + 5x + 3) 
3. Find the gradient of y = x² - 3x at x = 4.

Integration 
4. ∫ (2x) dx 
5. ∫ (5x² + 3) dx 
6. ∫(0→3) x dx

Applied 
7. A particle has velocity v = 4t. Find displacement from t = 0 to t = 5. 
8. Find turning points of f(x) = x³ - 6x.

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Summary

This introduction covered:
• limits 
• derivatives 
• gradients 
• turning points 
• integrals 
• areas under curves 
• real-world applications 

Calculus is the language of change — mastering it unlocks physics, engineering, chemistry, biology, and advanced mathematics.