Express $\frac{\mathrm{d}}{\mathrm{d}x}\left[ {{ an }^{-1}}(xy) ight]$ in terms of $x$, $y$ and $

Question

Express $\frac{\mathrm{d}}{\mathrm{d}x}\left[ {{	an }^{-1}}(xy) ight]$ in terms of $x$, $y$ and $\frac{\mathrm{d}y}{\mathrm{d}x}$. It is given that $x$ and $y$ satisfy the equation: ${{	an }^{-1}}x+{{	an }^{-1}}y+{{	an }^{-1}}(xy)=\frac{7}{12}\pi $. Find the value of $y$ when $x=1$. Show that, when $x=1$, $\frac{\mathrm{d}y}{\mathrm{d}x}=-\frac{1}{3}-\frac{1}{2\sqrt{3}}$.

Accepted Answer

(i) 1/(1+x²y²)(x dy/dx + y); (ii) x = 1, y = 1/√3; (iii) dy/dx = -1/3 - 1/(2√3)

Differentiation Involving Inverse Trigonometric Functions and Implicit Relations - AOD

Differentiation Involving Inverse Trigonometric Functions and Implicit Relations - AOD

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