Differentiation from First Principles

• The Derivative: The derivative, denoted as $f'(x)$ or $\frac{dy}{dx}$, represents the instantaneous rate of change or the gradient of the tangent to a curve at any given point $x$.

• The Limit Definition: Differentiation from first principles relies on the formal limit definition: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

• Variable 'h': In this context, $h$ represents an infinitesimal change in the $x$-coordinate, often referred to as $\delta x$ in some texts. It is the horizontal distance between two points on a curve used to calculate a chord's gradient.

• Chord to Tangent: The gradient of a chord between $(x, f(x))$ and $(x+h, f(x+h))$ is given by the change in $y$ divided by the change in $x$, which is $\frac{f(x+h) - f(x)}{h}$.

• The Limiting Process: As $h$ gets smaller and smaller ($h \to 0$), the two points on the curve move closer together, and the chord eventually becomes the tangent line at point $x$.

• Indeterminate Forms: If we simply set $h=0$ at the start, the formula results in $\frac{0}{0}$, which is undefined; therefore, algebraic manipulation is required to cancel the $h$ in the denominator before evaluating the limit.

Step-by-Step Methodology

• Step 1: Setup: Write down the function $f(x)$ and find the expression for $f(x+h)$ by replacing every $x$ in the original function with $(x+h)$.
• Step 2: Substitution: Substitute both $f(x)$ and $f(x+h)$ into the first principles formula: $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$.
• Step 3: Expansion: Expand the brackets in the numerator (e.g., using binomial expansion for powers) and simplify the expression.
• Step 4: Cancellation: Subtract $f(x)$ from $f(x+h)$; all terms that do not contain a factor of $h$ should cancel out at this stage.
• Step 5: Division: Factor out $h$ from the remaining terms in the numerator and divide by the $h$ in the denominator.
• Step 6: Limit Evaluation: Apply the limit by letting $h = 0$ in the remaining expression to find the final derivative.

• Notation Discipline: Always write $\lim_{h \to 0}$ at the start of every line until the very final step where $h$ is actually substituted with zero; failing to do so can result in lost marks.

• The 'No-h' Check: After subtracting $f(x)$ in the numerator, every remaining term MUST contain at least one $h$. If you have a constant or an $x$-term without an $h$, you have made an algebraic error in your expansion.

• Verification: Use standard differentiation rules (like the power rule) to mentally check what the answer should be before you start the long algebraic process.

• Brackets Matter: When substituting $f(x+h)$, always use brackets, especially if the function has a coefficient (e.g., $3(x+h)^2$) or multiple terms.

• Incorrect Expansion: A common error is expanding $(x+h)^2$ as $x^2 + h^2$ instead of the correct $x^2 + 2xh + h^2$.

• Sign Errors: When subtracting $f(x)$, students often forget to distribute the negative sign across all terms of the original function.

• Premature Limit: Attempting to 'plug in' $h=0$ before canceling the $h$ in the denominator, which leads to an undefined expression.