If y= x^2+a^2 , prvoe that y dy dx -x=0 - Vidyadip

Question

If $\text{y}=\sqrt{\text{x}^2+\text{a}^2},$ prvoe that $\text{y}\frac{\text{dy}}{\text{dx}}-\text{x}=0$

✓

Answer

Here, $\text{y}=\sqrt{\text{x}^2+\text{a}^2}$
Differentiating with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\sqrt{\text{x}^2+\text{a}^2}\big)$
$=\frac{1}{2\sqrt{\text{x}^2+\text{a}^2}}\frac{\text{d}}{\text{dx}}\big(\text{x}^2+\text{a}^2\big)$
[Using chain rule]
$=\frac{1}{2\sqrt{\text{x}^2+\text{a}^2}}\times(2\text{x})$
$=\frac{\text{x}}{\sqrt{\text{x}^2+\text{a}^2}}$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{x}}{\text{y}}\ \Big[\text{Since},\sqrt{\text{x}^2+\text{a}^2}=\text{y}\Big]$
$\Rightarrow \text{y}\frac{\text{dy}}{\text{dx}}=\text{x}$
$\Rightarrow\text{y}\frac{\text{dy}}{\text{dx}}-\text{x}=0$

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→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

A bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black.

Form the differential equation of the family of circle in the secound qudrant and touching the coordinate axes.

Solve the following differential equation:
$\text{2x}^{2}\frac{\text{dy}}{\text{dx}}-\text{2 xy + y}^{2}=0$

Find the equation of the curve which passes through the point (1, 2) and the distance between the foot of the ordinate of the point of contact and the point of intersection of the tangent with x-axis twice abscissa of the pont of contact.

Show that the vectors $2\hat{\text{i}}-3\hat{\text{j}}+4\hat{\text{k}}$ and $-4\hat{\text{i}}+6\hat{\text{j}}-8\hat{\text{k}}$ are collinear.

Find the point on the curve $\frac{\text{x}^2}{9}+\frac{\text{y}^2}{16}=1$ at which the tangent are:

parallel to x-axis
paralle to y- axis

A card is drawn and replaced in an ordinary pack of 52 cards. How many times must a card be drawn so that.

there is at least an even chance of drawing a heart.
the probability of drawing a heart is greater than $\frac{3}{4}$?

Evaluate the following definite integrals:
$\int_{0}^\limits{\frac{\pi}{6}}\cos\text{x }\cos2\text{x}\text{ dx}$

Find the shortest distance between the following lines whose vector equations are:$\overrightarrow{r}=\text{(1 - t)}\hat{\text{i}}+\text{(t - 2)}\hat{\text{j}}+\text{(3 - 2t)}\hat{\text{k}}$ and
$\overrightarrow{r}=\text{(s + 1)}\hat{\text{i}}+\text{(2s - 1)}\hat{\text{j}}-\text{(2s + 1)}\hat{\text{k}}$

Let C be the set of all complex numbers and $C_0$ be the set of all no-zero complex numbers. Let a relation R on $C_0$ be defined as $\text{z}_1\text{R z}_2\Leftrightarrow\frac{\text{z}_1-\text{z}_2}{\text{z}_1+\text{z}_2}$ is real for all $\text{z}_1,\ \text{z}_2\in\text{C}_0.$ Show that R is an equivalence relation.