Question

$\frac{d}{dx}\left(\frac{2^x+3^x}{4^x}\right)=$ ___________

• A. ${\left(\frac{1}{2}\right)}^x\log \frac{1}{2}-{\left(\frac{3}{2}\right)}^x\log \frac{3}{4}$
• B. ${\left(\frac{1}{2}\right)}^x\log \frac{1}{2}-{\left(\frac{3}{4}\right)}^x\log \frac{3}{4}$
• C. ${\left(\frac{1}{2}\right)}^x\log \frac{1}{2}+{\left(\frac{3}{4}\right)}^x\log \frac{3}{4}$
• D. ${\left(\frac{1}{2}\right)}^x\log \frac{1}{2}+{\left(\frac{3}{4}\right)}^x\log \frac{3}{2}$

Answer 1

Answer: (C)

${\left(\frac{1}{2}\right)}^x\log \frac{1}{2}+{\left(\frac{3}{4}\right)}^x\log \frac{3}{4}$

For differentiation of $a^x-$

Let $y=a^x⟹\log y=x\log a$

Differentiating both sides-

$\frac{1}{y}\frac{dy}{dx}=\log a$

$⟹\frac{d{a}^x}{dx}=a^x\log a$

For given question-

$\frac{d}{dx}\left(\frac{2^x+3^x}{4^x}\right)$

$=\frac{d}{dx}{\left(\frac{2}{4}\right)}^x+\frac{d}{dx}{\left(\frac{3}{4}\right)}^x$

$={\left(\frac{1}{2}\right)}^x\log \frac{1}{2}+{\left(\frac{3}{4}\right)}^x\log \frac{3}{4}$