Differentiation from First Principles

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Differentiation from First Principles

The derivative is $f'(x)=\lim_{h\to0}[f(x+h)-f(x)]/h$ . Expanding accurately and cancelling before taking the limit turns average change into instantaneous gradient.

Worked reasoning

$[3(x+h)^2-3x^2]/h=[6xh+3h^2]/h$ .
Simplify to $6x+3h$ .
Let $h\to0$ to obtain $6x$ .

Key idea

First principles derives the tangent gradient as the limit of secant gradients.

Use first principles or a known derivation to find the derivative of $f(x)=3x^2$ .

Which statement best captures the central mathematical idea in Differentiation from First Principles?

First principles derives the tangent gradient as the limit of secant gradients.

Substitute $h=0$ into the original quotient before simplifying.

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Differentiation from First Principles, which move is most reliable?

Write $f(x+h)$ explicitly before forming and simplifying the difference quotient.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Differentiation from First Principles?

Substitute $h=0$ into the original quotient before simplifying.

First principles derives the tangent gradient as the limit of secant gradients.

Write $f(x+h)$ explicitly before forming and simplifying the difference quotient.

Differentiate by a rule afterward and compare the result.

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