Question

$y=\frac{{\cos }^{2}x{.2}^{x^2}(x^3-2x+1)}{\tan x\cosh x.e^{x-2}}$, find $\frac{dy}{dx}$

• A. $\frac{dy}{dx}=y[-2\tan x+2x\log 2+\frac{3x^2-2}{x^3-2x+1}-\frac{{\sec }^{2}x}{\tan x}-\frac{\sinh x}{\cosh x}-1]$
• B. $\frac{dy}{dx}=y[2\tan x+2x\log 2+\frac{3x^2+2}{x^3-2x+1}-\frac{{\sec }^{2}x}{\tan x}+\frac{\sinh x}{\cosh x}-1]$
• C. $\frac{dy}{dx}=y[2\tan x+2x\log 2+\frac{3x^2+2}{x^3-2x+1}-\frac{{\sec }^{2}x}{\tan x}-\frac{\sinh x}{\cosh x}+1]$
• D. $\frac{dy}{dx}=y[-2\tan x+2x\log 2+\frac{3x^2-2}{x^3-2x+1}-\frac{{\sec }^{2}x}{\tan x}-\frac{\sinh x}{\cosh x}+1]$

Answer 1

The correct option is A $\frac{dy}{dx}=y[-2\tan x+2x\log 2+\frac{3x^2-2}{x^3-2x+1}-\frac{{\sec }^{2}x}{\tan x}-\frac{\sinh x}{\cosh x}-1]$

$y=\frac{{\cos }^{2}x{.2}^{x^2}(x^3-2x+1)}{\tan x\cosh x.e^{x-2}}$
Taking log,
$\log y=2\log \left(\cos x\right)+x^2\log 2+\log (x^3-2x+1)-\log \left(\tan x\right)-\log \left(\cosh x\right)-(x-2)$
Differentiating both side w.r.t $x$
$\frac{dy}{dx}=y[-2\tan x+2x\log 2+\frac{3x^2-2}{x^3-2x+1}-\frac{{\sec }^{2}x}{\tan x}-\frac{\sinh x}{\cosh x}-1]$