Question

Find $f^{\prime }\left(x\right)$
$f\left(x\right)=\sqrt{4^x-{\left(44\right)}^x}+\frac{3}{{\log }_4\left(1-x\right)}$

Answer 1

Given ,

$f\left(x\right)=\sqrt{4^x+{44}^x}+\frac{3}{{\log }_4(1-x)}$

Differentiate both side w.r. to x , we get

$f^{\prime }\left(x\right)=\frac{1}{2\sqrt{4^x+{44}^x}}\frac{d}{dx}(4^x+{44}^x)+3\frac{d}{dx}\frac{1}{{\log }_4(1-x)}$

$=\frac{1}{2\sqrt{4^x+{44}^x}}(4^x{\log }_{e}4+{44}^x{\log }_{e}44)+3(-1)\frac{1}{{\left({\log }_4(1-x)\right)}^2}\frac{d}{dx}{\log }_4(1-x)$

$=\frac{4^x{\log }_{e}4+{44}^x{\log }_{e}44}{2\sqrt{4^x+{44}^x}}-\frac{3}{{\left({\log }_4(1-x)\right)}^2}\times \frac{1}{(1-x){\log }_{e}4}\frac{d}{dx}(1-x)$

$=\frac{4^x{\log }_{e}4+{44}^x{\log }_{e}44}{2\sqrt{4^x+{44}^x}}-\frac{3}{{\left({\log }_4(1-x)\right)}^2}\times \frac{1}{(1-x){\log }_{e}4}(-1)$

$=\frac{4^x{\log }_{e}4+{44}^x{\log }_{e}44}{2\sqrt{4^x+{44}^x}}+\frac{3}{{\left({\log }_4(1-x)\right)}^2}\times \frac{1}{(1-x){\log }_{e}4}$