Question

Find the derivative of cos x from first principle.

Answer 1

Here f(x)= cos x

Then f(x+h) = cos(x+h)

We know that $f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{(x+h)-f\left(x\right)}{h}$

$\Rightarrow f^{\prime }\left(x\right)={lim}_{h\to 0}\frac{cos(x+h)-cosx}{h}$

$={lim}_{h\to 0}\frac{-2sin\left(\frac{2x+h}{2}\right)sin\left(\frac{h}{2}\right)}{h}$

$={lim}_{h\to 0}-sin\left(\frac{2x+h}{2}\right).\frac{sin\left(\frac{h}{2}\right)}{\left(\frac{h}{2}\right)}$

$=-sinx$