Gradients, Tangents and Normal
Finding the normal
To find the normal, differentiate your function and find the tangent. Then, negetive reciprocal your gradient to find the perpendicular line.
Find the equation of the normal to the curve \(y=x+3\sqrt{ x }\) when x = 9.
- Find y: \(y=9+3\sqrt{ 9 } \to y=18\)
- Find gradient: \(y=x+3\sqrt{ x } \to 1+\frac{3}{2x}^{-\frac{1}{2}}\)
- Sub in: \(1+\frac{3}{2}\left( 9^{-\frac{1}{2}} \right)=\frac{3}{2}\)
- Find normal gradient: \(\frac{3}{2} \to -\frac{2}{3}\)