If y=e^ x +e^ -x , prvoe that dy dx = y^2-4 - Vidyadip

Question

If $\text{y}=\text{e}^{\text{x}}+\text{e}^{-\text{x}},$ prvoe that $\frac{\text{dy}}{\text{dx}}=\sqrt{\text{y}^2-4}$

✓

Answer

Differentiating with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\big(\text{e}^{\text{x}}+\text{e}^{-\text{x}}\big)$
$=\frac{\text{d}}{\text{dx}}\text{e}^{\text{x}}+\frac{\text{d}}{\text{dx}}{\text{e}}^{-\text{x}}$
$=\text{e}^{\text{x}}+\text{e}^{-\text{x}}\frac{\text{d}}{\text{dx}}\big(-\text{x}\big)$
[Using chain rule]
$=\text{e}^{\text{x}}+\text{e}^{-\text{x}}(-1)$
$=\big(\text{e}^{\text{x}}-\text{e}^{-\text{x}}\big)$
$=\sqrt{\big(\text{e}^{\text{x}}-\text{e}^{-\text{x}}\big)^2-4\text{e}^{\text{x}}\times\text{e}^{-\text{x}}}$
$\Big[\text{Since},(\text{a}-\text{b}=\sqrt{(\text{a}+\text{b})^2-4\text{ab}}\Big]$
$=\sqrt{\text{y}^2-4}$
$\big[\text{Since e}^\text{x}+\text{e}^{-\text{x}}=\text{y}\big]$
Hence, the solution is, $\frac{\text{dy}}{\text{dx}}=\sqrt{\text{y}^2-4}$

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 CONTINUITY AND DIFFERENTIABILITY→CONTINUITY AND DIFFERENTIABILITY — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Evaluate the following integrals:
$\int\text{cosec x}\log({\text{cosec x}-\cot\text{x})}\text{dx}$

If $y^x + x^y + x^x = a^b$, find $\frac{\text{dy}}{\text{dx}}$

Integrate the following w.r.t. x
$\frac{x^{2} - 3x + 1}{\sqrt{1 - x^{2}}}$

$\int\bigg[\log \log \text{x} + \frac{1}{(\log \text{x)}^{2}}\bigg]\text{dx}$

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,
$\text{f}(\text{x})=\text{x}\sqrt{2-\text{x}^{2}}-\sqrt{2}\leq\text{x}\leq\sqrt{2}$

There are three urns A, B, and C. Urn A contains 4 red balls and 3 black balls. urn B contains 5 red balls and 4 black balls. Urn C contains 4 red and 4 black balls. One ball is drawn from each of these urns. What is the probability that 3 balls drawn consists of 2 red balls and a black ball?

Find the area of the region common to the parabolas $4y^2= 9x$ and $3x^2= 16y$.

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 $\text{f}\text{(x)}=\begin{cases}\text{e}^\frac{1}{\text {x}}, & \text{if} \text{ x}\neq 0\\1, & \text{if}\text{x} = 0\end{cases}$ find whethe f is continuous at x = 0.

If $\int_{0}^\limits{\text{k}}\frac{1}{2+8\text{x}^2}\text{ dx}=\frac{\pi}{16},$ find the value of k.