If `f(x)=|cosx|` , find `f^(prime)(pi/4)` and `f^(prime)((3pi)/4)` .

Explanation

You cannot directly differentiate `f(x)=|cos x|`. Take cases around `frac{pi}{4}` and `frac{3 pi}{4}`.


`x in[0, frac{pi}{2}) Rightarrow cos x>0 `

`Rightarrow|cos x|=cos x `

`Rightarrow f^{prime}(x)=frac{{d}}{{d} x}|cos x| `

`Rightarrow f^{prime}(x)=frac{{d}}{{d} x} cos x `

`Rightarrow f^{prime}(x)=-sin x `

`Rightarrow f^{prime}(frac{pi}{4})=-sin frac{pi}{4} `

...