Download App

Differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\text{a}+\text{b}\tan\text{x}}{\text{b}-\text{a}\tan\text{x}}\Big)$
✓

Answer

Let, $\text{y}=\tan^{-1}\Big[\frac{\text{a}+\text{b}\tan\text{x}}{\text{b}-\text{a}\tan\text{x}}\Big]$
$\Rightarrow\text{y}=\tan^{-1}\bigg[\frac{\frac{\text{a}+\text{b}\tan\text{x}}{\text{b}}}{\frac{\text{b}-\text{a}\tan\text{x}}{\text{b}}}\bigg]$
$\Rightarrow\text{y}=\tan^{-1}\bigg[\frac{\frac{\text{a}}{\text{b}}+\tan\text{x}}{1-\frac{\text{a}}{\text{b}}\tan\text{x}}\bigg]$
$\Rightarrow\text{y}=\tan^{-1}\Bigg[\frac{\tan\Big(\tan^{-1}\frac{\text{a}}{\text{b}}\Big)+\tan\text{x}}{1-\tan\Big(\tan^{-1}\frac{\text{a}}{\text{b}}\Big)\times\tan\text{x}}\Bigg]$
$\Rightarrow\text{y}=\tan^{-1}\Big[\tan\Big(\tan^{-1}\frac{\text{a}}{\text{b}}+\text{x}\Big)\Big]$
$\Rightarrow\text{y}=\tan^{-1}\big(\frac{\text{a}}{\text{b}}\big)+\text{x}$
Differentiate it with respect to x,
$\frac{\text{dy}}{\text{dx}}=0+1$
$\therefore\ \frac{\text{dy}}{\text{dx}}=1$

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 "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

If the matrix $\begin{bmatrix}0&\text{a}&3\\2&\text{b}&-1\\\text{c}&1&0\end{bmatrix}$ is a skew-symmetric matrix, then find the values of a, b and c.
→
State when the line $\vec{\text{r}}=\vec{\text{a}}+\lambda\vec{\text{b}}$ is parallel to the plane $\vec{\text{r}}\cdot\vec{\text{n}}=\text{d}.$ Show that the line $\vec{\text{r}}=\hat{\text{i}}+\hat{\text{j}}+\lambda(3\hat{\text{i}}-\hat{\text{j}}+2\hat{\text{k}})$ is parallel to the plane $\vec{\text{r}}\cdot(2\hat{\text{i}}+\hat{\text{k}})=3.$ Also, find the distance between the line and the plane.
→
Find the equation of the normal to the curve $ay^2 = x^3 $ at the point $(am^2, am^3).$
→
Solve the following systems of linear equations by cramer's rule:
2x - y = 17,
3x + 5y = 6
→
Solve the following differential equation:
$\frac{\text{dy}}{\text{dx}}=\text{y}\tan\text{x}-2\sin\text{x}$
→
Prove the following results:
$\tan^{-1}\frac{1}{4}+\tan^{-1}\frac{2}{9}=\frac{1}{2}\cos^{-1}\frac{3}{5}=\frac{1}{2}\sin^{-1}\Big(\frac{4}{5}\Big)$
→
Differentiate the following functions with respect to x:
$\text{e}^\text{x}\log\sin2\text{x}$
→
A function $f: R \rightarrow R$ is defined as $f(x)=x^3+4$. Is it a bijection or not? In case it is a bijection, find $f^{-1}(3)$.
→
Find the coordinates of the foot of the perpendicular from the point (1, 1, 2) to the plane 2x - 2y + 4z + 5 = 0. Also, find the length of the perpendicular.
→
If $\begin{vmatrix}\text{p}&\text{q}&\text{c}\\\text{a}&\text{q}&\text{c}\\\text{a}&\text{b}&\text{r} \end{vmatrix}=0,$ find the value of $\frac{\text{p}}{\text{p}-\text{a}}+\frac{\text{q}}{\text{q}-\text{b}}+\frac{\text{r}}{\text{r}-\text{c}},$ $\text{p}\neq\text{a},\text{q}\neq\text{b},\text{r}\neq\text{c}.$
→