- #1
efekwulsemmay
- 54
- 0
Homework Statement
[tex]y=sec^{-1}\frac{1}{t}, 0<t<1[/tex]
Homework Equations
[tex]\frac{d}{dx}sec^{-1} x= \frac{1}{\left|x\right|\cdot\sqrt{x^{2}-1}}[/tex]
The Attempt at a Solution
Basically to simplify things I used u substitution so I let u=1/t then du/dt=-1/t2 and I got:
[tex]y=sec^{-1}u\rightarrow y^{,}=\frac{1}{\left|u\right|\cdot\sqrt{u^{2}-1}}\cdot\frac{du}{dt}[/tex]
which,when I substituted for u, I got:
[tex]=\frac{1}{\left|\frac{1}{t}\right|\cdot\sqrt{\left(\frac{1}{t}\right)^{2}-1}}\cdot\frac{-1}{t^{2}}[/tex]
which works out as:
[tex]=\frac{-1}{\left|\frac{1}{t}\right|\cdot t^{2}\cdot\sqrt{\frac{1^{2}}{t^{2}}-1}}[/tex]
and:
[tex]=\frac{-1}{t\cdot\sqrt{1-t^{2}}}[/tex]
however, the answer as per the back of the book is:
[tex]\frac{-1}{\sqrt{1-t^{2}}}[/tex]
what did I do wrong?