Differentiation — common mistakes

A very common mistake is to use the tangent gradient when asked for the normal's equation. After you find $\frac{dy}{dx}$ at the point, immediately write down $m_{\text{tangent}} = ...$ and $m_{\text{normal}} = ...$ on your script. This simple step prevents you from mixing them up under pressure.

$\frac{\delta y}{\delta x}$ (using the Greek letter delta) represents a finite change in y divided by a finite change in x, like the gradient of a chord between two points. $\frac{dy}{dx}$ represents the limit of this ratio as the change in x becomes infinitesimally small. It is the true gradient of the curve at a single point, not an approximation between two points.

No. The Product Rule, Quotient Rule, and Chain Rule are part of the P3 (Pure Mathematics 3) syllabus. For P1, you are only expected to differentiate sums of terms of the form $ax^n$. If you see a product or quotient, you must first expand or divide it to get a sum of terms before differentiating.

A negative gradient means that the function is decreasing at that point. As you move from left to right along the x-axis, the y-value of the curve is going down. This corresponds to a tangent line that slopes downwards.

First, find the gradient function by differentiating to get $\frac{dy}{dx}$. Then, set your gradient function equal to the desired gradient. For example, solve the equation $\frac{dy}{dx} = 5$. The solution(s) for $x$ will give you the x-coordinate(s) where the curve has that gradient.