Question

Show that if $x=\tan u$, then $dx={\sec }^{2}udu$, in the following two ways: (a) by using implicit differentiation, thinking of $u$as a function of $x$, and (b) by thinking of $x$as a function of $u$.

Short answer

Ans: It is proved that $x=\tan u,\text{then}dx={\sec }^{2}udu$by (a) and (b) both ways.

Step-by-step solution

Step 1. given information.

given,

$x=\tan u,\text{then}dx={\sec }^{2}udu$

Step 2. (a)  The objective is to show that dx=sec2⁡udu by using implicit differentiation by assuming u as a function of x.

The differentiation is done by assuming $u$as a function of $x$.

The differentiation is shown below.

$\begin{array}{r}x=\tan u\\ 1={\sec }^{2}u\frac{du}{dx}\\ dx={\sec }^{2}udu\end{array}$

Hence the expression is proved.

Step 3. (b)  The objective is to show that dx=sec2⁡uduby using implicit differentiation by assuming x as a function of u.

The differentiation is done by assuming $x$as a function of $u$.

The differentiation is shown below.

$\begin{array}{r}x=\tan u\\ \frac{dx}{du}={\sec }^{2}u\\ dx={\sec }^{2}udu\end{array}$

Hence the expression is proved.