Derivative of f(x) = sin^2 (x) by limit definition

[Voelho]

Hello,
I'm having problem on this, and I hope that someone can give me some ideas to how solve it.
Since f(x) = sin^2 (x), it's derivative is sin(2x), or using the trigonometric identity, sin(2x) = 2*sinx*cosx.
But I'm getting nowhere by the limit definition, the furthest I could had ended in the limit: Lim (h->0) of cos²(x) - cos²(x + h) / h , using a lot of Trigonometric Identities on the way...

The start limit is Lim (h->0) of Sin²(x + h) - Sin²(x) / h

Any ideas?

 

[HallsofIvy]

Are you required to use the limit definition? The whole point of developing the "chain rule" is to avoid such a complicated calculation! If you are required to do this "the hard way", then you will probably want to combine the techniques used for the derivatives of \(\displaystyle x^2\) and sin(x). That is, \(\displaystyle sin^2(x+ h)- sin^2(x)= (sin(x+h)+ sin(x))(sin(x+h)- sin(x))\). Taking the limit of "sin(x+h)+ sin(x)" as h goes to 0 is easy- it is, of course, sin(x)+ sin(x)= 2sin(x). The important part is the remaining \(\displaystyle \frac{sin(x+h)- sin(x)}{h}\). Follow the derivative of sin(x) to do that.

 

[pka]

Since f(x) = sin^2 (x), it's derivative is sin(2x), or using the trigonometric identity, sin(2x) = 2*sinx*cosx. But I'm getting nowhere by the limit definition, the furthest I could had ended in the limit: Lim (h->0) of cos²(x) - cos²(x + h) / h , using a lot of Trigonometric Identities on the way...
The start limit is Lim (h->0) of Sin²(x + h) - Sin²(x) / h

Factor as the difference of two squares:

\(\displaystyle \dfrac{{{{\sin }^2}(x + h) - {{\sin }^2}(x)}}{h} = \dfrac{{\sin (x + h) - \sin (x)}}{h}(\sin (x + h) + \sin (x))\)

\(\displaystyle \begin{align*} \dfrac{{\sin (x + h) - \sin (x)}}{h} &= \dfrac{{\sin (x)\cos(h)+\cos(x)\sin(h) - \sin (x)}}{h}\\&=\sin(x)\left[\dfrac{cos(h)-1}{h}\right] +\cos(x)\dfrac{\sin(h)}{h}\end{align*}\)

 

[Voelho]

Thanks, I finally made it. I used the differences of two squares and isolated the derivative of sin(x).
The question asks to do it through "the hard way", I think it is to learn these steps and then pass to the chain rule method.