Download App

Differentiation — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Find $\frac{\text{dy}}{\text{dx}},$ when
$\text{x}=\cos^{-1}\frac{1}{\sqrt{1+\text{t}^2}}\text{ and y}=\sin^{-1}\frac{\text{t}}{\sqrt{1+\text{t}^2}},\text{t}\in\text{R}$
✓

Answer

We have, $\text{x}=\cos^{-1}\Big(\frac{1}{\sqrt{1+\text{t}^{2}}}\Big)$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\frac{-1}{\sqrt{1-\Big(\frac{1}{\sqrt{1+\text{t}^{2}}}\Big)^{2}}}\frac{\text{d}}{\text{dt}}\Big(\frac{1}{\sqrt{1+\text{t}}^{2}}\Big)$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\frac{-1}{\sqrt{1-\frac{1}{(1+\text{t}^{2})}}}\left\{\frac{-1}{2(1+\text{t}^{2})^\frac{3}{2}}\right\}\frac{\text{d}}{\text{dt}}(1+\text{t}^{2})$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\frac{(1+\text{t}^{2})^{\frac{1}{2}}}{\sqrt{1+\text{t}^{2}-1}}\times\frac{1}{2(1+\text{t}^{2})^\frac{3}{2}}(2\text{t})$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\frac{\text{t}}{\sqrt{\text{t}^{2}}\times(1+\text{t}^{2})}$
$\Rightarrow\frac{\text{dx}}{\text{dt}}=\frac{1}{1+\text{t}^{2}}...(\text{i})$
Now, $\text{y}=\sin^{-1}\Big(\frac{1}{\sqrt{1+\text{t}^{2}}}\Big)$
$\Rightarrow\frac{\text{dy}}{\text{dt}}=\frac{1}{\sqrt{1-\Big(\frac{1}{\sqrt{1+\text{t}^{2}}}\Big)^{2}}}\frac{\text{d}}{\text{dt}}\Big(\frac{1}{\sqrt{1+\text{t}^{2}}}\Big)$
$\Rightarrow\frac{\text{dy}}{\text{dt}}=\frac{1}{\sqrt{1-\frac{1}{(1+\text{t}^{2})}}}\left\{\frac{-1}{2(1+\text{t}^{2})\frac{3}{2}}\right\}\frac{\text{d}}{\text{dt}}(1+\text{t}^{2})$
$\Rightarrow\frac{\text{dy}}{\text{dt}}=\frac{(1+\text{t}^{2})^{\frac{1}{2}}}{\sqrt{1+\text{t}^{2}-1}}\times\frac{-1}{2(1+\text{t}^{2})^{\frac{3}{2}}}(2\text{t})$
$\Rightarrow\frac{\text{dy}}{\text{dt}}=\frac{-1}{2\sqrt{\text{t}^{2}}\times(1+\text{t}^{2})}(2\text{t})$
$\Rightarrow\frac{\text{dy}}{\text{dt}}=\frac{-1}{1+\text{t}^{2}}....(\text{ii})$
Dividing equation (ii) by (i),
$\frac{\frac{\text{dy}}{\text{dt}}}{\frac{\text{dx}}{\text{dt}}}=\frac{1}{(1+\text{t}^{2})}\times\frac{(1+\text{t}^{2})}{-1}$
$\Rightarrow\frac{\text{dy}}{\text{dx}}=-1$

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

Show that among all positive number $x$ and $y$ with $x^2+y^2=r^2$ the sum $x+y$ is largest when $x=y=\sqrt{2}$.
→
Solve the following differential equation
$\text{x}\cos\text{y dy}=(\text{xe}^\text{x}\log\text{x}+\text{e}^\text{x})\text{dx}$
→
If f'(x) = a sin x + b cos x and f'(0) = 4, f(0) = 3, $\text{f}\Big(\frac{\pi}{2}\Big)=5$, find f(x).
→
Let $\text{F}(\alpha)=\begin{bmatrix}\cos\alpha & -\sin\alpha & 0 \\ \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 1\end{bmatrix}$$\text{and G }(\beta)=\begin{bmatrix} \cos\beta & 0 & \sin\beta \\ 0 & 1 & 0 \\ -\sin\beta & 0 & \cos\beta \end{bmatrix}$
Show that$\big[\text{F}(\alpha)\text{G}(\beta)\big]^{-1}=\text{G}(-\beta)\text{F}(-\alpha).$
→
Solve the following differential equations:$\text{y}(1+\text{e}^{\text{x}})\text{dy}=(\text{y}+1)\text{e}^{\text{x}}\text{ dx}$
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big\{\frac{\text{x}}{\sqrt{\text{a}^2-\text{x}^2}}\Big\},-\text{a}<\text{x}<\text{a}$
→
Solve the following differential equation
$(\sin\text{x}+\cos\text{x})\text{dy}+(\cos\text{x}+\sin\text{x})\text{dx}=0$
→
A company sells two different products, A and B. The two products are produced in a common production process, which has a total capacity of 500 man-hours. It takes 5 hours to produce a unit of A and 3 hours to produce a unit of B. The market has been surveyed and company officials feel that the maximum number of unit of A that can be sold is 70 and that for B is 125. If the profit is Rs. 20 per unit for the product A and Rs. 15 per unit for the product B, how many units of each product should be sold to maximize profit?
→
Show that the lines $\frac{\text{x}-1}{3}=\frac{\text{y}+1}{2}=\frac{\text{z}-1}{5}$ and $\frac{\text{x}+2}{4}=\frac{\text{y}-1}{3}=\frac{\text{z}+1}{-2}$ do not intersect.
→
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{2^{\text{x}+1}}{1-4^{\text{x}}}\Big),-\infty<\text{x}<0$
→