Question

The slope of the tangent to the curve $y=\ln \left(\cos x\right)$ at $x=\frac{3\pi }{4}$ is

• A. $1$
• B. $-1$
• C. $\ell n\sqrt{2}$
• D. $\frac{1}{\sqrt{2}}$

Answer 1

Answer: (A)

$1$

$y=\ln \left(\cos x\right)$

Differentiating w.r.t. $x,$

$\Rightarrow$ $\frac{dy}{dx}=\frac{1}{\cos x}\times (-\sin x)$

$\Rightarrow$ $\frac{dy}{dx}=\frac{-\sin x}{\cos x}$

$\Rightarrow$ $\frac{dy}{dx}=-\tan x$

Since, $x=\frac{3\pi }{4}$ means $x={135}^o$

$\Rightarrow$ $\frac{dy}{dx}=-\tan \left({135}^o\right)$

$\Rightarrow$ $\frac{dy}{dx}=-\tan ({90}^o+{45}^o)$

$\Rightarrow$ $\frac{dy}{dx}=-(-\cot {45}^o)$ [ Since, $\tan ({90}^o+\theta )=-\cot \theta$ ]

$\Rightarrow$ $\frac{dy}{dx}=-(-1)$

$\Rightarrow$ $\frac{dy}{dx}=1$

$\Rightarrow$ Slope of the tangent $=\frac{dy}{dx}=1$