Question

If the value of $y = \log \left[ {\tan x} \right],$ find$\frac{{dy}}{{dx}}$.

Answer 1

Hint: you can differentiate it by using formulae of logarithm or
Using substitution method.

We have to differentiate it with respect to $x$.
We will use a substitution method.
Let, $\tan x = t$
$\begin{gathered} \therefore y = \log t \\\ dy = \frac{1}{t}dt \\\ \end{gathered}$
$\frac{{dy}}{{dt}} = \frac{1}{t} = \frac{1}{{\tan x}}$ $\ldots \ldots \left( 1 \right)$
Now
$\begin{gathered} t = \tan x \\\ dt = {\sec ^2}xdx \\\ \frac{{dt}}{{dx}} = {\sec ^2}x \ldots \ldots \left( 2 \right) \\\ \end{gathered}$
On multiplying equation $\left( 1 \right)$and $\left( 2 \right)$
We get,
$\begin{gathered} \frac{{dy}}{{dt}} \times \frac{{dt}}{{dx}} = \frac{{{{\sec }^2}x}}{{\tan x}} \\\ \frac{{dy}}{{dx}} = \frac{{\cos x}}{{\cos x.\cos x.\sin x}} = \frac{1}{{\sin x.\cos x}} \\\ \frac{{dy}}{{dx}} = \frac{1}{{\sin x.\cos x}} \\\ \end{gathered}$
Is the required answer.

Note:Using a substitution method you can solve easily. You have to find
Differentiation with respect to $x$, then you can differentiate it with respect
to $t$ and further differentiate $t$ with respect to $x$