Question

differentiate w.r.t x

Answer 1

Please cite your sources @Rohangrr as lgba said

Answer 2

To differentiate an expression: \(\Large x^n\), the formula is: \(\Large nx^{n - 1}\)

Answer 3

(tan^-1){[(sqrt)(1+((a^2)(x^2)))]-1}/ax
differentiate w.r.t x

Answer 4

is it:
\[\tan^{-1}(\frac{\sqrt{x}(1+x^{2}x^{2})-1}{ax}) \]?

Answer 5

\[\tan^{-1}(\frac{\sqrt{1+a^2x^2}-1}{ax})\]???

Answer 6

yea thats it

Answer 7

\[\frac{d}{dx}(\tan^{-1} x)=\frac{1}{1+x^2}\]
Using that here
\[\frac{d}{dx} \tan^{-1} (\frac{\sqrt{1+a^2x^2}-1}{ax})=\frac{1}{1+(\frac{\sqrt{1+a^2x^2}-1}{ax})^2}\times \frac{d}{dx} (\frac{\sqrt{1+a^2x^2}-1}{ax})\]
\[=\frac{1}{1+(\frac{\sqrt{1+a^2x^2}-1}{ax})^2}\times (\frac{ax \times \frac{d}{dx}(\sqrt{1+a^2x^2}-1)-(\sqrt{1+a^2x^2}-1)\frac{d}{dx}(ax)}{a^2x^2})\]
Do you understand till here?
after this only simplification is needed

Answer 8

\[\small =\frac{1}{1+(\frac{\sqrt{1+a^2x^2}-1}{ax})^2}\times (\frac{ax \times \frac{d}{dx}(\sqrt{1+a^2x^2}-1)-(\sqrt{1+a^2x^2}-1)\frac{d}{dx}(ax)}{a^2x^2})\]

Answer 9

i tried it be substituting x=tanx/a
but after 3 steps dint no wat to do
could you try substituting and then differentiationg it

Answer 10

Ok, I'll do that

Answer 11

yea thanks

Answer 12

Oh no, the page restarted.
Wait I'll type again

Answer 13

okay

Answer 14

Let's substitute
\[x=\frac{\tan u}{a}\]
\[y=\tan^{-1} (\frac{\sqrt{1+a^2x^2}-1}{ax})\]
After substitution, we get
\[y=\tan^{-1} (\frac{\sqrt {\sec^2u}-1}{\tan u})\]

Answer 15

Do you get this part?

Answer 16

yea

Answer 17

We know
\[1-\cos u= 2\sin^2 \frac u 2\]
and
\[\sin u= 2\sin \frac u2 \cos \frac u2\]
Let's substitute these, we get
\[y=\tan^{-1} (\frac{2 \sin^2 \frac u 2}{2\sin \frac u2 \cos \frac u2})\]
So we get
\[y=\tan ^{-1} ({\tan \frac u2})\]

Answer 18

So we get
\[y=\frac u 2
\]
or
\[\frac{dy}{du}=\frac 12\]
We had substituted
\[x=\frac {\tan u}{a}\]
or
\[u= \tan ^{-1} ax\]
so
\[\frac{du}{dx}= a \times \frac{1}{1+a^2x^2}\]
I think you can do now, can't you?

Answer 19

yea thanks

Answer 20

Phew !!
The page restarted so many times:)