Download App

Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
If $\text{y}=\log\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\},$ show that $\frac{\text{dy}}{\text{dt}}=\frac{-1}{2\sqrt{\text{x}^2-1}}.$
✓

Answer

Here $\text{y}=\log\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\}$
Differentiating it with respect to x and applying the chain and product rule, we get
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}} \log\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\}$
$\frac{\text{dy}}{\text{dx}}=\frac{1}{\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\}}\frac{\text{d}}{\text{dx}}(\sqrt{\text{x}-1}-\sqrt{\text{x}+1})$
$=\frac{1}{\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\}}\Big[\frac{\text{d}}{\text{dx}}\sqrt{\text{x}-1}-\frac{\text{d}}{\text{dx}}\sqrt{\text{x}+1}\Big]$
$=\frac{1}{\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\}}\Big[\frac{1}{2}(\text{x}-1)^{\frac{1}{2}}-\frac{1}{2}(\text{x}+1)^{\frac{1}{2}}\Big]$
$=\frac{1}{2}\frac{1}{\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\}}\Big[\frac{1}{\sqrt{\text{x}-1}}-\frac{1}{\sqrt{\text{x}+1}}\Big]$
$=\frac{1}{2}\frac{1}{\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}+1}\big\}}\Bigg(\frac{-\big\{\sqrt{\text{x}-1}-\sqrt{\text{x}-1}\big\}}{\big(\sqrt{\text{x}-1}\big)\big(\sqrt{\text{x}+1}\big)}\Bigg)$
$=\frac{1}{2}\bigg(\frac{1}{\big(\sqrt{\text{x}-1}\big)\big(\sqrt{\text{x}+1}\big)}\bigg)$
$\frac{\text{dy}}{\text{dx}}=\frac{-1}{2\sqrt{\text{x}^2-1}}$
Therefore,
$\frac{\text{dy}}{\text{dx}}=\frac{-1}{2\sqrt{\text{x}^2-1}}$

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 "Solve the Following Question.(5 Marks)" from Differentiation→Differentiation — all question types→Maths STD 12 Science question papers→Maharashtra Board STD 12 Science question papers→Maharashtra Board question paper generator→

Similar questions

By computing the shortest distance, determine whether following lines intersect each other.

$\bar{r}=(\hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{k})$ and $\bar{r}=(2 \hat{i}-\hat{j})+\mu(\hat{i}+\hat{j}-\hat{k})$

→
Differentiate $\sin^{-1}\Big(2\text{x}\sqrt{1-\text{x}^2}\Big)$ with respect to $\sec^{-1}\Big(\frac{1}{\sqrt{1+\text{x}^2}}\Big),$ if:
$\text{x}\in\Big(0,\frac{1}{\sqrt{2}}\Big)$
→
Evaluate the following integrals:$\int\frac{2\text{x}}{2+\text{x}-\text{x}^2}\text{ dx}$
→
Show that the following curves intersect orthogonally at the indicated points:
$y^2 = 8x $and $2x^2 + y^2 = 10 $at $\big(1,2\sqrt{2})$
→
Find an equation for the set all points that are equidistant from the planes $3x - 4y + 12z = 6$ and $4x + 3z = 7$
→
Solve the following systems of linear equations by cramer's rule:
$3x + y + z = 2,$
$2x - 4y + 3z = -1,$
$4x + y - 3z = -11$
→
Let A be the set of all human beings in a town at a particular time. Determine whether the following relations are reflexive, symmetric and transitive:
R = {(x, y): x is wife of y}
→
Differential equation $\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0,\text{y}(0)=1,\text{y}'(0)=1$Function $\text{y}=\sin\text{x}+\cos\text{x}$
→
Evaluate the following integrals:$\int\text{e}^{\text{x}}\frac{(1-\text{x})^2}{(1+\text{x}^2)^2}\text{dx}$
→
Evaluate the following intregals:
$\int\frac{1}{5+7\cos\text{x}+\sin\text{x}}\ \text{dx}$
→