Gradients, tangents and normals

Finding the gradient at a point

• The gradient function $\dfrac{dy}{dx}$ gives the gradient at any point, so:
  • Differentiate the curve to get $\dfrac{dy}{dx}$.
  • Substitute the $x$-value of the point to get the exact gradient there.

What a tangent is

• A tangent is a straight line that touches the curve at one point and has the same gradient as the curve at that point.
• Its equation is found with the point–gradient form of a line:

$y - y_1 = m(x - x_1)$

• $m$ is the gradient at the point (from the derivative).
• $(x_1, y_1)$ is the point of contact (from the original curve).
• Method: find the point's $y$-coordinate from the curve, find the gradient $m$ from the derivative, then substitute both into $y - y_1 = m(x - x_1)$.

What a normal is

• A normal is the line perpendicular (at right angles) to the tangent at the same point.
• Perpendicular gradients multiply to $-1$, so the normal's gradient is the negative reciprocal of the tangent's — flip it and change the sign.

Increasing, decreasing and stationary

• Where $\dfrac{dy}{dx} > 0$ the curve is increasing (sloping up).
• Where $\dfrac{dy}{dx} < 0$ the curve is decreasing (sloping down).
• Where $\dfrac{dy}{dx} = 0$ the curve is momentarily flat — a stationary point.