APPLICATION OF DERIVATIVES — MATHS STD 12 Science — Question

1. A vertical tangent (parallel to y-axis).

Solution:

We are given that $\text{y}=\text{x}^{\frac{1}{5}}$

$\Rightarrow\ \frac{\text{dy}}{\text{dx}}\Rightarrow\frac{\text{dy}}{\text{dx}}=\frac{1}{5}\text{x}^{\frac{1}{3}-1}$ $\Big[\because\frac{\text{d}}{\text{dx}}(\text{x}^\text{n})=\text{nx}^{\text{n}-1}\Big]$

$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{1}{5}\text{x}^{\frac{-4}{5}}$

$\Rightarrow\ \frac{\text{dy}}{\text{dx}}=\frac{1}{5\text{x}^{\frac{4}{5}}}$

$\Rightarrow\ \Big(\frac{\text{dy}}{\text{dx}}\Big)_{(0,0)}=\frac{1}{5(0)^{\frac{4}{5}}}=\infty$

So, the curve $\text{y}=\text{x}^{\frac{1}{5}}$ has a vertical tangent at (0, 0), which is parallel to Y-axis.