Question

If y = tan-1(6x-7/6+7x)
then dy/dx is
​

Answer 1

$\textbf{Given:}$

$\mathsf{y=tan^{-1}\left(\dfrac{6x-7}{6+7x}\right)}$

$\textbf{To find:}$

$\mathsf{\dfrac{dy}{dx}}$

$\textbf{Solution:}$

$\textsf{We apply chain rule to find derivative of the given function}$

$\mathsf{Consider,}$

$\mathsf{y=tan^{-1}\left(\dfrac{6x-7}{6+7x}\right)}$

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

$\mathsf{\dfrac{dy}{dx}=\dfrac{1}{1+\left(\dfrac{6x-7}{6+7x}\right)^2}\;\left(\dfrac{(6+7x)6-(6x-7)7}{(6+7x)^2}\right)}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{(6+7x)^2}{(6+7x)^2+(6x-7)^2}\;\left(\dfrac{36+42x-42x+49}{(6+7x)^2}\right)}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{(6+7x)^2}{(6+7x)^2+(6x-7)^2}\;\left(\dfrac{85}{(6+7x)^2}\right)}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{85}{(6+7x)^2+(6x-7)^2}}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{85}{36+49x^2+84x+36x^2+49-84x}}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{85}{36+49x^2+36x^2+49}}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{85}{85+85x^2}}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{85}{85(1+x^2)}}$

$\implies\boxed{\mathsf{\dfrac{dy}{dx}=\dfrac{1}{1+x^2}}}$

$\textbf{Find more:}$

If y=e^2x (ax+b), then prove that d^2 y/dx^2 -4 dy/dx+4y=0

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