Definition of Derivatives

The derivative is geometrically defined as the gradient of the tangent to a curve at a specific point. While a chord (secant line) measures the average rate of change between two points, a tangent measures the rate at exactly one point.

To find the actual gradient at point $x$, we consider a second point very close to it, denoted as $x + h$. The gradient of the chord joining these two points is given by the formula $\frac{f(x+h) - f(x)}{h}$.

As the distance $h$ between the two points approaches zero, the chord becomes the tangent line. This transition is mathematically captured using the concept of a limit.

The formal definition of a derivative, often called 'differentiation from first principles', is expressed as: $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$

Step 1: Identify the function $f(x)$ and determine the expression for $f(x+h)$ by substituting $(x+h)$ into every instance of $x$.

Step 2: Substitute these into the difference quotient $\frac{f(x+h) - f(x)}{h}$ and simplify the numerator to cancel out terms.

Step 3: Divide by $h$ and then evaluate the limit by letting $h$ approach zero to find the final derivative expression.

Conceptual Understanding: In many exams, you are not required to derive derivatives from first principles for every problem, but you must understand that the derivative is the limit of the chord gradient.

Visual Checks: Always check if a curve is 'smooth' at the point of interest. If a graph has a sharp corner (like a modulus function), the tangent is not uniquely defined, and the derivative does not exist at that point.

Notation Awareness: Be comfortable switching between $f'(x)$ and $\frac{dy}{dx}$. Use $f'(x)$ when dealing with function notation and $\frac{dy}{dx}$ when working with equations in terms of $y$ and $x$.

Division by Zero: A common mistake is trying to set $h=0$ before simplifying the expression. The limit process requires algebraic simplification to remove $h$ from the denominator first.

Confusing Coordinates with Gradients: Students often mistake the $y$-coordinate of a point for the gradient at that point. The $y$-value tells you 'where' the point is, while the derivative tells you 'how fast' it is moving.

Linear vs. Non-linear: Remember that for a linear function $y=mx+c$, the derivative is constant ($m$). For non-linear functions, the derivative is usually a function of $x$, meaning the gradient changes as you move along the curve.

The definition of the derivative is the foundation for all differentiation rules, such as the Power Rule and Chain Rule. These rules are simply shortcuts derived from the first principles limit formula.

In physics, the derivative of a displacement-time graph represents velocity, and the derivative of a velocity-time graph represents acceleration, illustrating the real-world application of rates of change.