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

Last updated date: 26th Mar 2023

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Answer

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310.2k+ views

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 $x$

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 $x$

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