I need help on the following:
1)Differentiate with respect with x:
I got , however i don't understand why the book's answer is .
2)Differentiate with respect with x:
This is what i have done:
What would i do next?
3) Prove that
How would i approach this??
3. Just in case a picture helps...
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
The general drift is...
Don't integrate - balloontegrate!
Balloon Calculus; standard integrals, derivatives and methods
Balloon Calculus Drawing with LaTeX and Asymptote!
OK, in this version, you start with y = arccos x, top left.
Substitute (or map to) cos theta for x, then differentiate with respect to theta using the chain rule. And this is equal to differentiating theta with respect to theta. So...
So, using Pythag,
Then back-substitute x for cos theta.
A different way is to start with
Then differentiate with respect to x...
Substitute arccos x for y...
Then use a right-triangle diagram or look up at http://staff.jccc.net/swilson/trig/compositions.htm to simplify.