Question

$f\left(x\right)={\int }_{e^x}^{e^{3x}}\frac{tdt}{\ln t,}x>0$ find differential coefficient of f(x) w.r.t. $\ln x$ when $x=\ln 2$

• A. 60
• B. 40
• C. 50
• D. 30

Answer 1

Answer: (A)

60

Let $y=f\left(x\right)={\int }_{e^x}^{e^{3x}}\frac{tdt}{\log t}$
$\frac{dy}{dx}=\frac{e^{3x}}{\log e^{3x}}\frac{d\left(e^{3x}\right)}{dx}-\frac{e^x}{\log e^x}\frac{d\left(e^x\right)}{dx}$
$\frac{dy}{dx}=\frac{3e^{6x}}{3x}-\frac{e^{2x}}{x}$
$\frac{dy}{dx}=\frac{e^{6x}-e^{2x}}{x}$
Now, let $t=\log x$
$\frac{dt}{dx}=\frac{1}{x}$
$\frac{dy}{dt}=e^{6x}-e^{2x}$
$\Rightarrow {\left(\frac{dy}{dt}\right)}_{x=\log 2}=2^6-2^2=64-4=60$