# Finding the Normal Equation

A normal to a curve is a straight line passing through the point where the tangent touches the curve and is perpendicular (at right angles) to the tangent at that point. The gradient of the tangent to a curve is $m$, then the gradient of the normal is $\displaystyle -\dfrac{1}{m}$, as the product of the gradients of $2$ perpendicular lines equals to $-1$.
The gradient of the tangent at $x=a$ is $f'(a)$. Therefore the gradient of the normal is $\displaystyle -\dfrac{1}{f'(a)}$. The equation of the normal is: $$y-f(a) = -\dfrac{1}{f'(a)}(x-a)$$

### Example 1

Find the gradient of the normal to the curve $f(x)=2x^3-x^2+1$ at $x=1$.

### Example 2

Find the equation of the normal to $f(x)=x^3-2x+3$ at the point $(1,2)$.

### Example 3

Find the equation of the normal to $f(x)=3x^2-1$ at $x=1$.