# Help Pleas (d/dx)

• Sep 10th 2008, 06:57 PM
tir0nel
Basic Calculas (d/dx)
ok (d/dx) sin x= cos x

therefore (d/dx)cos x= -sinx

but how do u explain in terms how (d/dx)cosx = sin x

i believe i am supposed to us the formula f'(x)= F(x + h) - f(x) / h

but can someone show me how it is done so i can understand how it is expressed.
• Sep 10th 2008, 07:30 PM
Jameson
Ok. So the limit definition by your notation should be something like $\displaystyle f'(x)=\lim_{h \rightarrow 0} \frac{ \cos{(x+h)}-\cos{x}}{h}$. So to handle the numerator use this identity - $\displaystyle \cos{(a+b)}=\cos{(a)}\cos{(b)}-\sin{(a)}\sin{(b)}$. Apply this to your limit and post back what you find. (Wink)
• Sep 10th 2008, 07:41 PM
Jameson
Thinking more that actually is a tricky limit. Try just using the fact that $\displaystyle \sin{ \left( \frac{\pi}{2} + x \right)} = \cos{x}$ and that $\displaystyle \sin(a+b) = \sin(a) \sin(b) + \cos(a) \cos(b)$
• Sep 10th 2008, 07:54 PM
11rdc11
$\displaystyle f'(x)=\lim_{h \rightarrow 0} \frac{ \cos{(x+h)}-\cos{x}}{h}$

$\displaystyle \lim_{h \rightarrow 0}\frac{cos(x)cos(h)-sin(x)sin(h) - cos(x)}{h}$

$\displaystyle \lim_{h \rightarrow 0}\frac{-sin(x)sin(h) -cos(x)(1- cos(h))}{h}$

$\displaystyle -\lim_{h \rightarrow 0}\frac{sin(x)sin(h)}{h} -\lim_{h \rightarrow 0}\frac{cos(x)(1-cos(h))}{h}$

$\displaystyle -sin(x)\lim_{h \rightarrow 0} \frac{sin(h)}{h}-cos(x)\lim_{h \rightarrow 0}\frac{1-cos(h)}{h}$

Then remember that

$\displaystyle \lim_{h \rightarrow 0} \frac{sin(h)}{h} = 1$

and

$\displaystyle \lim_{h \rightarrow 0} \frac{1-cos(h)}{h}=0$

You should be able to finish up from here and that was a tricky question lol
• Sep 10th 2008, 09:55 PM
Jameson
^ Well just do it all why don't you? (Rofl)

Nicely done. OP that's exactly what you want.
• Sep 10th 2008, 10:03 PM
11rdc11
lol yep I was stuck for a good bit becuause the cos(x) in the numerator would cancel out leaving $\displaystyle \frac{0}{0}$
• Sep 10th 2008, 10:10 PM
Moo
Yop !

A more straighforward method is to use $\displaystyle \cos(a)-\cos(b)=-2 \sin \frac{a+b}{2} \sin \frac{a-b}{2}$

$\displaystyle \frac{\cos(x+h)-\cos(x)}{h}=-2 \cdot \frac{\sin(x+\tfrac h2) \sin(\tfrac h2)}{h}=-2 \cdot \sin(x+\tfrac h2) \cdot \frac{\sin(\tfrac h2)}{h}$

We know that $\displaystyle \lim_{x \to 0} \frac{\sin(ax)}{x}=a.$ Therefore $\displaystyle \lim_{h \to 0} \frac{\sin(\tfrac h2)}{h}=\frac 12$

And we have $\displaystyle \lim_{h \to 0} \frac{\cos(x+h)-\cos(x)}{h}=-\sin(x)$