# How do you implicitly differentiate csc(x^2/y^2)=e^(xy) ?

Sep 2, 2017

dy/dx=(2xcsc(x^2/y^2)cot(x^2/y^2)+y^3e^(xy))/(xy(2xcsc(x^2/y^2)cot(x^2/y^2)-ye^(xy))

#### Explanation:

Implicit differentiation is no different from explicit differentiation. Just remember that differentiating a function of $y$ causes the chain rule to be in effect. Recall also that $\frac{d}{\mathrm{dx}} \csc x = - \csc x \cot x$ and $\frac{d}{\mathrm{dx}} {e}^{x} = {e}^{x}$.

$\csc \left({x}^{2} / {y}^{2}\right) = {e}^{x y}$

$\frac{d}{\mathrm{dx}} \csc \left({x}^{2} / {y}^{2}\right) = \frac{d}{\mathrm{dx}} {e}^{x y}$

$- \csc \left({x}^{2} / {y}^{2}\right) \cot \left({x}^{2} / {y}^{2}\right) \cdot \frac{d}{\mathrm{dx}} \left({x}^{2} {y}^{-} 2\right) = {e}^{x y} \cdot \frac{d}{\mathrm{dx}} \left(x y\right)$

Use the product rule to find these derivatives. Recall that while $\frac{d}{\mathrm{dx}} {x}^{2} = 2 x$, $\frac{d}{\mathrm{dx}} {y}^{2} = 2 y \frac{\mathrm{dy}}{\mathrm{dx}}$.

$- \csc \left({x}^{2} / {y}^{2}\right) \cot \left({x}^{2} / {y}^{2}\right) \left(2 x {y}^{-} 2 - 2 {x}^{2} {y}^{-} 1 \frac{\mathrm{dy}}{\mathrm{dx}}\right) = {e}^{x y} \left(y + x \frac{\mathrm{dy}}{\mathrm{dx}}\right)$

Expanding and rearranging to group $\frac{\mathrm{dy}}{\mathrm{dx}}$ terms:

$\left(\frac{2 {x}^{2} \csc \left({x}^{2} / {y}^{2}\right) \cot \left({x}^{2} / {y}^{2}\right)}{y} - x {e}^{x y}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x \csc \left({x}^{2} / {y}^{2}\right) \cot \left({x}^{2} / {y}^{2}\right)}{y} ^ 2 + y {e}^{x y}$

Common denominators:

$\left(\frac{2 {x}^{2} \csc \left({x}^{2} / {y}^{2}\right) \cot \left({x}^{2} / {y}^{2}\right) - x y {e}^{x y}}{y}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x \csc \left({x}^{2} / {y}^{2}\right) \cot \left({x}^{2} / {y}^{2}\right) + {y}^{3} {e}^{x y}}{y} ^ 2$

Solving for $\frac{\mathrm{dy}}{\mathrm{dx}}$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x \csc \left({x}^{2} / {y}^{2}\right) \cot \left({x}^{2} / {y}^{2}\right) + {y}^{3} {e}^{x y}}{y} ^ 2 \cdot \frac{y}{2 {x}^{2} \csc \left({x}^{2} / {y}^{2}\right) \cot \left({x}^{2} / {y}^{2}\right) - x y {e}^{x y}}$

dy/dx=(2xcsc(x^2/y^2)cot(x^2/y^2)+y^3e^(xy))/(xy(2xcsc(x^2/y^2)cot(x^2/y^2)-ye^(xy))