Derivatives and the Gradient Function

The gradient of a straight line is its change in $y$ divided by its change in $x$. A curve can become steeper or flatter, so its gradient depends on where you measure it.

Use two points on $y=x^2$: $P=(1,1)$ and $Q=(2,4)$. The straight line joining them is a chord. Its gradient is

This describes the change between two points, not the gradient at $P$ alone. Keep $P$ fixed and move $Q$ closer. The chord approaches the tangent: the straight line with the curve's direction at $P$.

In the graph, $a$ is the x-coordinate of $P$, and $h$ is the horizontal change from $P$ to $Q$. Start with $a=1$ and reduce positive $h$. Then approach from the left using negative $h$.

The value approached from both sides is called the limit and gives the tangent gradient at the chosen $P$. We write $h\to0$ for “$h$ approaches zero”. We do not divide by zero or calculate a chord through two identical points.

Differentiation finds a gradient rule, called the derivative. For $y=f(x)=x^2$, the rule is $f'(x)=2x$. Moving $P$ in the graph shows how the gradient changes.

$f'(x)$, read “f prime of x”, and $\dfrac{dy}{dx}$ both describe the derivative. “With respect to $x$” means that $x$ is the variable being changed. This notation does not mean $y/x$.

At $x=3$, $f(3)=9$ is the height, but $f'(3)=6$ is the gradient. A derivative need not exist at every point: a sharp corner has no single tangent direction.