Differentiate the following functions with respect to x: ^ -1 ( 1… - Vidyadip

Question

Differentiate the following functions with respect to x:
$\tan^{-1}\Big(\frac{\sqrt{1+\text{a}^2\text{x}^2-1}}{\text{ax}}\Big),\text{x}\neq0$

✓

Answer

Let $\text{y}=\tan^{-1}\Big(\frac{\sqrt{1+\text{a}^2\text{x}^2}-1}{\text{ax}}\Big)$
Put $\text{ax}=\tan\theta$
$\text{y}=\tan^{-1}\Big(\frac{\sqrt{1+\text{a}^2\text{x}^2}-1}{\text{ax}}\Big)$
$=\tan^{-1}\Big(\frac{\sec\theta-1}{\tan\theta}\Big)$
$=\tan^{-1}\Big(\frac{1-\cos\theta}{\sin\theta}\Big)$
$=\tan^{-1}\bigg(\frac{\frac{2\sin^2\theta}{2}}{\frac{2\sin\theta}{2}\frac{\cos\theta}{2}}\bigg)$
$\text{y}=\tan^{-1}\Big(\frac{\tan\theta}{2}\Big)$
$=\frac{\theta}{2}$
$\text{y}=\frac{1}{2}\tan^{-1}(\text{ax})$
Differentiating it with respect to x using chain rule,
$\frac{\text{dy}}{\text{dx}}=\frac{1}{2}\times\Big(\frac{1}{1+(\text{ax})^2}\Big)\frac{\text{d}}{\text{dx}}(\text{ax})$
$\frac{\text{dy}}{\text{dx}}=\frac{1}{2(1+\text{a}^2\text{x}^2)}(\text{a})$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{a}}{2(1+\text{a}^2\text{x}^2)}$

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

Find the points of local maxima or local minima and corresponding local maximum and local minimum values of the following functions. Also, find the points of inflection,$f'(x) = x^4- 62x^2 + 9x + 15$

A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are-as follows:

Machine	Area occupied	Labour force	Daily output ( in units)
A	$1000m^2$	12 men	60
B	$1200m^2$	8 men	40

He has maximum area of $9000 \text{m}^{2}$ available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximise the daily output?

Let $A$ and $B$ be two sets. Show that $f : A \times B \rightarrow B \times A$ such that $f ( a , b )=( b , a )$ is
$(i) $injective
$(ii)$ bijective

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).

If $\text{x}=\text{a}\sin2\text{t}(1+\cos 2\text{t})$ and $\text{y}=\text{b}\cos\text{t}(1-\cos2\text{t}),$ show that at $\text{t}=\frac{\pi}{4},\frac{\text{dy}}{\text{dx}}=\frac{\text{b}}{\text{a}}\text{ t}=\frac{\pi}{4},\frac{\text{dy}}{\text{dx}}=\frac{\text{b}}{\text{a}}$

Find the equations of the tangent and the normal to the following curves at the indicated points.
$4\text{x}^2+9\text{y}^2=36\text{ at }(3\cos\theta,2\sin\theta)$

Find all points of discontinuity of $f,$ where $f$ is defined by:
$\text{f(x)}= \begin{cases}\text{x}^3 - 3,\ \ \text{if x}\leq 2 \\\text{x}^2 + 1,\ \text{if x}>2\end{cases}$

If A and B are two events such that $2\text{P(A)}=\text{P(B)}=\frac{5}{13}$ and $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big)=\frac{2}{5}$ find $\text{P}(\text{A}\cap\text{B}).$

Find the shortest distance between the lines
$\vec{\text{r}}=\big(\hat{\text{i}}+2\hat{\text{j}}+\hat{\text{k}}\big)+\lambda\big(\hat{\text{i}}-\hat{\text{j}}+\hat{\text{k}}\big)$ and, $\vec{\text{r}}=2\hat{\text{i}}-\hat{\text{j}}-\hat{\text{k}}+\mu\big(2\hat{\text{i}}+\hat{\text{j}}+2\hat{\text{k}}\big)$

Evaluate the following intregals:
$\int\frac{2\text{x}^2+7\text{x}-3}{\text{x}^2(2\text{x}+1)}\text{ dx}$