Question

Differentiate the following with respect to x

Answer 1

a) ln (sec nx)

Answer 2

@ganeshie8

Answer 3

chain rule again

Answer 4

\(u = \sec (nx)\)

\[ \begin{align}\dfrac{d}{dx}\ln (u) &= \dfrac{d}{du} \ln (u) \cdot \dfrac{d}{dx} u \\~\\ &= ?\end{align}\]

Answer 5

ln(u) = 1/sec(nx) ??

Answer 6

@ganeshie8

Answer 7

lets replace u's in the end okay ?

Answer 8

\[\dfrac{d}{du} \ln (u) = \dfrac{1}{u}\]

it says that the rate of change of function \(\ln (u)\) with respect to \(u\) is \(\frac{1}{u}\)

Answer 9

\(u = \sec (nx)\)

\[ \begin{align}\dfrac{d}{dx}\ln (u) &= \dfrac{d}{du} \ln (u) \cdot \dfrac{d}{dx} u \\~\\ &= \dfrac{1}{u} \cdot \dfrac{d}{dx} u \end{align}\]

Answer 10

Next replace u by sec(nx) :

\(u = \sec (nx)\)

\[ \begin{align}\dfrac{d}{dx}\ln (u) &= \dfrac{d}{du} \ln (u) \cdot \dfrac{d}{dx} u \\~\\ &= \dfrac{1}{u} \cdot \dfrac{d}{dx} u\\~\\
&= \dfrac{1}{\sec(nx)} \cdot \dfrac{d}{dx} \sec(nx)\\~\\

\end{align}\]

Answer 11

remember the derivative of sec(u) ?

Answer 12

No

Answer 13

sec(nx) = cosec nx ?

Answer 14

use this table for derivatives of few well known functions
http://www.math.com/tables/derivatives/tableof.htm

Answer 15

ok

Answer 16

So, is it sec(nx)= sec(nx) tan (nx) ?
@ganeshie8