If x^m + y^n = 1, Prove that dy dx =- my nx - Vidyadip

Question

If $x^m + y^n = 1,$ Prove that $\frac{\text{dy}}{\text{dx}}=-\frac{\text{my}}{\text{nx}}$

✓

Answer

We have, $x^m + y^n = 1$
Taking $\log $ on both side,
$\log(\text{x}^\text{m}\text{y}^{\text{n}})=\log(1)$
Differentiating with respect to $x,$
$\frac{\text{dy}}{\text{dx}}(\text{m}\log\text{x})+\frac{\text{d}}{\text{dx}}(\text{n}\log\text{y})=\frac{\text{d}}{\text{dx}}\big\{\log(1)\big\}$
$\Rightarrow\frac{\text{m}}{\text{n}}+\frac{\text{n}}{\text{y}}\frac{\text{dy}}{\text{dx}}=0$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-\frac{\text{m}}{\text{x}}\times\frac{\text{y}}{\text{n}}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-\frac{\text{my}}{\text{nx}}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from Differentiation→Differentiation — all question types→MATHS STD 12 Science question papers→Rajasthan Board STD 12 Science question papers→Rajasthan Board question paper generator→

Similar questions

Differentiate $\tan^{-1}\Big(\frac{2\text{x}}{1-\text{x}^2}\Big)$ with respect to $\cos^{-1}\Big(\frac{1-\text{x}^2}{1+\text{x}^2}\Big),$ if 0 < x < 1.

Evaluate the following integrals: $\int\frac{(\text{x}^2+1)(\text{x}^2+2)}{(\text{x}^2+3)(\text{x}^2+4)}\ \text{dx}$

Evaluate the following integrals:
$\int\cos\Big\{2\cot^{-1}\sqrt{\frac{1+\text{x}}{1-\text{x}}}\Big\}\text{dx}$

If $\text{x}^\text{y}-\text{y}^\text{x}=\text{a}^\text{b},$ find $=\frac{\text{dy}}{\text{dx}}.$

Find the length and the foot ofo perpendicular from the point $\Big(1,\frac{3}{2},2\Big)$ to the plane 2x - 2y + 4z + 5 = 0

Prove that:
$\begin{vmatrix}(\text{b}+\text{c})^2&\text{a}^2&\text{bc}\\\text{c}+\text{a})^2&\text{b}^2&\text{ca} \\\text{a}+\text{b})^2&\text{c}^2&\text{ab}\end{vmatrix}$ $=(\text{a}-\text{b})(\text{b}-\text{c})(\text{c}-\text{b})(\text{a}+\text{b}+\text{c})(\text{a}^2+\text{b}^2+\text{c}^2)$

Find the area bounded by the curve y = sin x between x = 0 and x = 2n.

Show that the differential equation of $xdy - ydx = \sqrt { x ^ { 2 } + y ^ { 2 } } d x$, is homogeneous and solve it.

$\text{If x = a} (\cos 2\text{t +2t}\sin \text{2t}) \text{and y = a} (\sin \text{2t - 2t}\cos\text{2t}),\text{then find}\frac{\text{d}^{2}{\text{y}}}{\text{dx}^{2}}.$

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{a}+\text{bx}}{\text{b}-\text{ax}}\Big)$