Quick questions on Tangents and normals to a curve explained: O-Level A-Maths

The gradient of the tangent at a point is the value of the derivative there. So:

The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of the tangent gradient:

Use the same point with the normal gradient in the point-gradient form to write the normal line.

A question may ask where a tangent is parallel to a given line, or horizontal. A parallel tangent has the same gradient as that line, so set the derivative equal to that gradient and solve for $x$; a horizontal tangent has gradient zero, the condition for a stationary point.

The normal gradient is the negative reciprocal; $\dfrac{1}{m}$ alone is wrong.

For $y = \dfrac{4}{x} = 4x^{-1}$, the derivative is $-4x^{-2}$; mind the negative power.

Find the tangent gradient to $y = x^3$ at $x = 1$. [2 marks]

Find the equation of the tangent to $y = x^2 + 1$ at $(1, 2)$. [3 marks]

State the normal gradient to a curve where the tangent gradient is $\dfrac{1}{4}$. [1 mark]