derivative trigonometry

**jvignacio** · September 11th 2008, 08:51 PM

$ f(x) = tanxsinx $

$ f '(x) = (sec^2x)(sinx) + (tanx)(cosx) $

this correcT?

**11rdc11** · September 11th 2008, 08:58 PM

Yep

$ f '(x) = (sec^2x)(sinx) + (tanx)(cosx) $

You could simplify it a little more

$tan(x)(secx+cosx)$

**jvignacio** · September 11th 2008, 09:00 PM

Originally Posted by 11rdc11

Yep

can we simplify this ?
if so how would u do it?

**Chris L T521** · September 11th 2008, 09:02 PM

Originally Posted by jvignacio

$ f(x) = tanxsinx $

$ f '(x) = (sec^2x)(sinx) + (tanx)(cosx) $

this correcT?

Yes, but it can be simplified further:

Note that $\sec^2x\sin x=\frac{1}{\cos^2x}\sin x=\tan x \sec x$ and $\tan x\cos x=\frac{\sin x}{\cos x}\cos x=\sin x$

So we now see that $f '(x) = (\sec^2x)(\sin x) + (\tan x)(\cos x)=\sec x\tan x+\sin x$

--Chris

**jvignacio** · September 11th 2008, 09:07 PM

Originally Posted by Chris L T521

Yes, but it can be simplified further:

Note that $\sec^2x\sin x=\frac{1}{\cos^2x}\sin x=\tan x \sec x$ and $\tan x\cos x=\frac{\sin x}{\cos x}\cos x=\sin x$

So we now see that $f '(x) = (\sec^2x)(\sin x) + (\tan x)(\cos x)=\sec x\tan x+\sin x$

--Chris

what if the possible answers were:

$ sinx - sinxsec^2x - sinx(sec^2x+1) - secx - none of these $

what would you go for?

**11rdc11** · September 11th 2008, 09:16 PM

I did this kind of fast but I'm pretty sure it the 3rd option

$ f '(x) =\sec x\tan x+\sin x $

$\frac{sinx}{cos^2x} + sinx$

$\frac{sinx+sinxcos^2x}{cos^2x}$

$\frac{sinx(1+cos^2x)}{cos^2x}$

$(sinx)(sec^2x + 1)$

**jvignacio** · September 11th 2008, 09:27 PM

Originally Posted by 11rdc11

I did this kind of fast but I'm pretty sure it the 3rd option

yeah thats what i put but i wasnt sure why we got rid of the tanx and cosx

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