## anonymous one year ago derivative of tan x using first principle

1. amistre64

dunno if it helps to use the sin/cos definintion

2. amistre64

i can never recall the tan(a+b) formula .. can you?

3. anonymous

i used quotient rule and got sec^2 x which i know is right but the first principle rule is troubling

4. anonymous

|dw:1433521302532:dw|

5. amistre64

wondering if this works any simpler | sin(x)cos(h) + sin(h)cos(x) sin(x) | 1 | ---------------------- - ----- | --- | cos(x)cos(h) - sin(x)sin(h) cos(x) | h

6. amistre64

$\frac{sin(x)}{cos(x+h)}\frac{cos(h) }{h}+\frac{cos(x)}{cos(x+h)}\frac{ sin(h) }{h}-\frac{sin(x)}{h~cos(x)}$

7. amistre64

do you recall what the cos(h) and sin(h) parts limit to?

8. anonymous

9. anonymous

i need to follow in similar manner

10. amistre64

i though you needed to work it by first principles so far ive got it to sin(x)cos(x) cos(h) + cos^2(x)sin(h) -sin(x)[cos(x)cos(h) - sin(x)sin(h)] ----------------------------------------------------------- h(cos(x+h)cos(x)) which allows us to get the h parts lined up

11. anonymous

then apply limits

12. amistre64

$\frac{\frac{sin(x+h)}{cos(x+h)}-\frac{sin(x)}{cos(x)}}{h}$ $\frac{cos(x)sin(x+h)-sin(x)cos(x+h)}{h~cos(x)cos(x+h)}$ $\frac{cos(x)}{cos(x)cos(x+h)} \frac{sin(x)sin(h)+cos(x)sin(h)}{h} \\ -\frac{sin(x)}{cos(x)cos(x+h)}\frac{cos(x)cos(h)-sin(x)sin(h)}{h}$

13. amistre64

got a typo ... ***sin(x)cos(h)*** in the expansion of sin(x+h)

14. amistre64

i cant see an error other than that .... but im biased. how is your tan version coming along?

15. anonymous

16. anonymous

if there is any mistake please point it out

17. amistre64

i dont see that you dropped the denominator of the top, down to the bottom the GCD of cos(x) and cos(x+h), is cos(x)cos(x+h)

18. amistre64

i recall that sin(h)/h limits to 1 is there a similar 'known' for cos(h)/h ?

19. anonymous

thank you ill try and correct the mistake :)

20. amistre64

[cos(h)-1]/h is what we need to deal with the cosine.

21. amistre64

for some recollection ... f' of sin(x) sin(x+h) - sin(x) ----------------- h sin(x)cos(h) +cos(x)sin(h) - sin(x) ----------------------------------- h sin(x)[cos(h)-1] +cos(x)sin(h) ------------------------------- h sin(x)[cos(h)-1]/h +cos(x)sin(h)/h sin(x)(0) +cos(x)(1) cos(x)

22. amistre64

so im sure theres some more factoring to do in order to approach our goal