y^(xsiny) +x^(ysinx) =1 determine (dy/dx)


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Question Number 49845 by mhozhez last updated on 11/Dec/18

$y^{x s i n y} + x^{y s i n x} = 1$ $d e t e r m i n e \frac{d y}{d x}$
Commented by MJS last updated on 11/Dec/18

$try this for yourself . the method is simple :$ $t_{i} (x, y) are terms in x and y$ $t_{1} (x, y) + t_{2} (x, y) + . . . = 0 is the same as$ $\underset{i = 1}{\sum^{n}} t_{i} (x, y) = 0$ $now derivate$ $d x \underset{i = 1}{\sum^{n}} \frac{d}{d x} [t_{i} (x, y)] + d y \underset{i = 1}{\sum^{n}} \frac{d}{d y} [t_{i} (x, y)] = 0$ $\Rightarrow \frac{d y}{d x} = - \frac{\underset{i = 1}{\sum^{n}} \frac{d}{d x} [t_{i} (x, y)]}{\underset{i = 1}{\sum^{n}} \frac{d}{d y} [t_{i} (x, y)]}$ $an example$ ${a x}^{2} + b x y + {c y}^{2} + d x + e y + f = 0$ $d x (2 a x + b y + 0 + d + 0 + 0) + d y (0 + b x + 2 c y + 0 + e + 0) = 0$ $d x (2 a x + b y + d) + d y (b x + 2 c y + e) = 0$ $\frac{d y}{d x} = - \frac{2 a x + b y + d}{b x + 2 c y + e}$
	Answered by tanmay.chaudhury50@gmail.com last updated on 11/Dec/18

	$u = y^{x s i n y}$ $l n u = x s i n y l n y$ $\frac{1}{u} \times \frac{d u}{d x} = x s i n y \times \frac{1}{y} \frac{d y}{d x} + x l n y \times c o s y \times \frac{d y}{d x} + s i n y l n y$ $\frac{d u}{d x} = y^{x s i n y} [\frac{d y}{d x} (\frac{x s i n y}{y} + x c o s y l n y) + s i n y l n y]$ $v = x^{y s i n x}$ $l n v = y s i n x l n x$ $\frac{1}{v} \frac{d v}{d x} = \frac{y s i n x}{x} + y l n x c o s x + s i n x l n x \frac{d y}{d x}$ $\frac{d v}{d x} = x^{y s i n x} (\frac{y s i n x}{x} + y l n x c o s x + s i n x l n x \frac{d y}{d x})$ $u + v = 1$ $\frac{d u}{d x} + \frac{d v}{d x} = 0$ $y^{x s i n y} [\frac{d y}{d x} (\frac{x s i n y}{y} + x c o s y l n y) + s i n y l n y] + x^{y s i n x} [\frac{y s i n x}{x} + y l n x c o s x + s i n x l n x \frac{d y}{d x}] = 0$ $\frac{d y}{d x} [y^{x s i n y} (\frac{x s i n y}{y} + x c o s y l n y) + (x^{y s i n x} s i n x l n x)] = - [x^{y s i n x} (\frac{y s i n x}{x} + y l n x c o s x) + y^{x s i n y} (s i n y l n y)]$ $\frac{d y}{d x} = \frac{- [x^{y s i n x} (\frac{y s i n x}{x} + y l n x c o s x) + y^{x s i n y} (s i n y l n y)]}{[y^{x s i n y} (\frac{x s i n y}{y} + x c o s y l n y) + (x^{y s i n x} s i n x l n x)]}$ $i t i s t h e a n s w e r . . .$
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