Question

Find out the value of derivative of log((10cosx).(5sinx)) at x = */ 4
5. A. Ooo
B. o
Coo
D. 01​

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{log[(10\,cosx)(5\,sinx)]}$

$\underline{\textbf{To find:}}$

$\textsf{Derivative of}$

$\mathsf{log[(10\,cosx)(5\,sinx)]\;at\;x=\dfrac{\pi}{4}}$

$\underline{\textbf{Solution:}}$

$\mathsf{Let\;y=log[(10\,cosx)(5\,sinx)]}$

$\textsf{This can be written as}$

$\mathsf{y=log(10\,cosx)+log(5\,sinx)}$

$\textsf{Differentiate with respect to 'x'}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{1}{10\,cosx}(-10\,sinx)+\dfrac{1}{5\,sinx}(5\,cosx)}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{-sinx}{cosx}+\dfrac{cosx}{sinx}}$

$\mathsf{\dfrac{dy}{dx}=-tanx+cotx}$

$\mathsf{Now,}$

$\mathsf{\left(\dfrac{dy}{dx}\right)_{\frac{\pi}{4}}}$

$\mathsf{=-tan\dfrac{\pi}{4}+cot\dfrac{\pi}{4}}$

$\mathsf{=-1+1}$

$\mathsf{=0}$

$\underline{\textbf{Answer:}}$

$\textsf{Option\;(B)\,is\,correct}$

$\underline{\textbf{Find more:}}$

If y = tanθ, then dy/dθ at θ = π/4 will be

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