## anonymous 5 years ago if y = (sin x)^x^2 then differentiate this!!!!! Pls give full steps!!!!!!

1. anonymous

chane rule

2. anonymous

use d f(x)^g(x)/ dx = f(x)^g(x)d/dx(g(x)log(f(x)) This can be proved easily

3. anonymous

=cos x *2x

4. anonymous

so here i gotta apply the chain rule got it so i have to learn it keep answering ppl i will just go through and then maybe I will be able to understand the solution!!!!

5. anonymous

keep posting what you do!

6. anonymous

ok

7. anonymous

@ruba: not you, I was telling that to tejeshwar so that he may understand and we may correct him if you are wrong

8. anonymous

Tejeshwar: this is not exaclty chain rule, f(x)^g(x) is a bit different thing

9. anonymous

nope sure its chain rule

10. anonymous

First take y=sin(x)^x^2 log(y)=x^2log(sinx) d/dx on both sides so you get, 1/y dy/dx = d/dx (x^2log(sinx)) therefore, dy/dx=y d/dx(x^2log(sin(x))

11. anonymous

$sin(x)^{x^2}$?

12. anonymous

ohhhhhhh so so so s o sry mogh im jst 4geting this

13. anonymous

if so take the log, simplify, take the derivative, then multiply by $sin(x)^{x^2}$

14. anonymous

ri8 ri8 sat & mogh go on nd bye

15. anonymous

Take this into consideration, y = f(x)^g(x) log y = g(x) log(f(x)) $(1/y)dy/dx = d/dx(g(x)\log(f(x)))$

16. anonymous

$ln(sin(x)^{x^2})=x^2ln(sin(x))$ $\frac{d}{dx}x^2ln(sin(x))=2xln(sin(x))+\frac{x^2}{x}=2xln(sin(x))+x$ answer: $(sin(x))^{x^2}(2xln(sin(x))+x)$

17. anonymous

oops sorry wrong

18. anonymous

$\frac{d}{dx}x^2ln(sin(x))=2xln(sin(x))+\frac{x^2}{sin(x)}$

19. anonymous

my mistake. sorry.

20. anonymous

@satellite: d/dx ln(sin(x)) is $cosx/sinx$

21. anonymous

wow am i off . you are right and i am wrong!

22. anonymous

two mistakes in one post i should resign.

23. anonymous

lol, it happens!

24. anonymous

btw it is also correct two write $(sin(x))^{x^2}=e^{x^2ln(sin(x))}$ and differentiate this using the chain rule. the actual work is identical since it all boils down to finding the derivative of $x^2ln(sin(x))$CORRECTLY

25. anonymous

just hang on satellite i will just be back 1ce i understand the chain rule!!!!

26. anonymous

Chain rule is easier than it looks, just check out the proof too! Regards.