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Updated Mar 25, 2026

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3 pages

Understanding Differentiation from First Principles

elle

@elle.xox

Differentiation from first principles is the fundamental method for finding... Show more

1 / 3

differentiating from first
principles
This is a graph of y = x² or f(x) = x²

f(x+h)
*B
A
x
Xth

The Gradient of Line AB = rise
nun
As B get

Understanding First Principles

Ever wondered how we actually work out the gradient of a curve at any point? First principles gives you the mathematical foundation behind all derivative calculations.

The key idea is brilliant in its simplicity: if you draw a line between two points on a curve and gradually move those points closer together, the line approaches the tangent to the curve. This tangent's gradient is your derivative.

The first principles formula captures this perfectly: f'(x) = lim[h→0] $f(x+h) - f(x)$ /h. Here, h represents the tiny distance between your two points, and as h approaches zero, you get the exact gradient.

Quick Tip: Think of first principles as finding the "instantaneous rate of change" - like working out your exact speed at one specific moment rather than your average speed over a journey.

Working Through Examples

Let's tackle some differentiation from first principles examples that show the method in action. These calculations might look intimidating at first, but there's a clear pattern you can follow every time.

For f(x) = 2x² + 3x, you substitute into the formula and get a fraction with algebra in both the numerator and denominator. The clever bit comes when you expand the brackets - this creates terms that cancel out, leaving you with expressions containing h.

After simplifying, you can factor out h from the numerator, which cancels with the h in the denominator. Once h disappears from the bottom, you can safely substitute h = 0 to get your final answer: f'(x) = 4x + 3.

Key Insight: The cancelling of h terms isn't just mathematical trickery - it's what allows the limit to exist and gives you a meaningful derivative.

More Complex Applications

First principles works brilliantly for higher powers and proves why our standard derivative rules actually make sense. When you differentiate x³, the algebra gets a bit messier, but the same systematic approach works.

The expansion of $x+h$ ³ uses the binomial theorem, giving you x³ + 3x²h + 3xh² + h³. After subtracting the original x³ and dividing by h, you're left with 3x² + 3xh + h².

As h approaches zero, those extra terms disappear, leaving you with f'(x) = 3x². This proves the power rule you've probably memorised - but now you understand why it works!

Pro Tip: Once you've mastered first principles, you'll appreciate how derivative rules like the power rule are actually shortcuts that save you from doing these lengthy calculations every time.

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•

Updated Mar 25, 2026

•

3 pages

Understanding Differentiation from First Principles

elle

@elle.xox

Differentiation from first principles is the fundamental method for finding derivatives - it shows you exactly how to calculate the rate of change of any function. Instead of memorising derivative rules, you'll learn the core logic behind how derivatives actually... Show more

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Understanding First Principles

Ever wondered how we actually work out the gradient of a curve at any point? First principles gives you the mathematical foundation behind all derivative calculations.

Quick Tip: Think of first principles as finding the "instantaneous rate of change" - like working out your exact speed at one specific moment rather than your average speed over a journey.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

Working Through Examples

Key Insight: The cancelling of h terms isn't just mathematical trickery - it's what allows the limit to exist and gives you a meaningful derivative.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

More Complex Applications

The expansion of $x+h$ ³ uses the binomial theorem, giving you x³ + 3x²h + 3xh² + h³. After subtracting the original x³ and dividing by h, you're left with 3x² + 3xh + h².

As h approaches zero, those extra terms disappear, leaving you with f'(x) = 3x². This proves the power rule you've probably memorised - but now you understand why it works!

Pro Tip: Once you've mastered first principles, you'll appreciate how derivative rules like the power rule are actually shortcuts that save you from doing these lengthy calculations every time.