Name: Derivative of Functions in Parametric Form
Uploaded: 2019-07-19
Duration: 8 min 37 s
Description: Learn everything you'll need to know about derivatives of functions in parametric forms with this quick and easy lesson that'll cover the concept comprehensi...

Question

If

y^{x} + x^{y} + x^{x} = a^{b}

, find

\frac{d y}{d x}

Accepted Answer

Given that $y^{x} + x^{y} + x^{x} = a^{b}$
Putting $u = y^{x}, v = x^{y}$ and $w = x^{x}$ , we get
$u + v + w = a^{b}$
$∴$ $\frac{d u}{d x} + \frac{d v}{d x} + \frac{d w}{d x} = 0 . . . (i)$
Now, $u = y^{x}$
Taking log on both sides,
$\log u = x \log y$
$\Rightarrow \frac{1}{u} \frac{d u}{d x} = x \frac{d}{d x} (\log y) + \log y \frac{d}{d x} (x)$ [using product rule]
$\Rightarrow \frac{1}{u} \frac{d u}{d x} = x \frac{1}{y} \frac{d y}{d x} + \log y \times 1$
$\Rightarrow \frac{d u}{d x} = u (\frac{x}{y} \frac{d y}{d x} + \log y)$
$\Rightarrow \frac{d u}{d x} = y^{x} (\frac{x}{y} \frac{d y}{d x} + \log y) . . . (ii)$
$v = x^{y}$
Taking log on both sides,
$\log v = y \log x$
$\Rightarrow \frac{1}{v} \frac{d v}{d x} = y \frac{d}{d x} (\log x) + \log x \frac{d y}{d x}$
$\Rightarrow \frac{1}{v} \frac{d v}{d x} = y \frac{1}{x} + \log x \frac{d y}{d x}$
$\Rightarrow \frac{d v}{d x} = v [\frac{y}{x} + \log x \frac{d y}{d x}]$
$\Rightarrow \frac{d v}{d x} = x^{y} [\frac{y}{x} + \log x \frac{d y}{d x}] . . . (iii)$
Again $w = x^{x}$
Taking log on both sides,
$\log w = x \log x$
$\Rightarrow \frac{1}{w} \frac{d w}{d x} = x \frac{d}{d x} (\log x) + \log x \frac{d}{d x} (x)$
$\Rightarrow \frac{1}{w} \frac{d w}{d x} = x \frac{1}{x} + \log x (1)$
$\Rightarrow \frac{d w}{d x} = w (1 + \log x)$
$\Rightarrow \frac{d w}{d x} = x^{x} (1 + \log x) . . . (iv)$
From $(i), (ii), (iii)$ and $(iv)$ we have
$y^{x} (\frac{x}{y} \frac{d y}{d x} + \log y) + x^{y} (\frac{y}{x} + \log x \frac{d y}{d x}) + x^{x} (1 + \log x) = 0$
$\Rightarrow (x . y^{x - 1} + x^{y} . \log x) \frac{d y}{d x} = - x^{x} (1 + \log x) - y . x^{y - 1} - y^{x} \log y$
$∴$ $\frac{d y}{d x} = \frac{- \{y^{x} \log y + y . x^{y - 1} + x^{x} (1 + \log x)\}}{x . y^{x - 1} + x^{y} \log x}$

If $y^{x} + x^{y} + x^{x} = a^{b}$ , find $\frac{d y}{d x}$

Important Questions on Differentiation

Learn and Prepare for any exam you want !

If yx+xy+xx=ab , find dydx

Important Questions on Differentiation

Learn and Prepare for any exam you want !

If $y^{x} + x^{y} + x^{x} = a^{b}$ , find $\frac{d y}{d x}$