Download App

Differentiation — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferentiation5 Marks
Question
Differentiate the following functions with respect to x:
$\frac{\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}}{\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}}$
✓

Answer

Let, $\text{y}=\frac{\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}}{\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}}$
Differentiate with respect to x,
$\frac{\text{dy}}{\text{dx}}=\frac{\text{d}}{\text{dx}}\Big[\frac{\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}}{\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}}\Big]$
$=\Bigg[\frac{\big(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}\big)\frac{\text{d}}{\text{dx}}\big(\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}\big)-\big(\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}\big)\frac{\text{d}}{\text{dx}}\big(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}\big)}{\big(\text{e}^{2\text{x}}-\text{e}^{4\text{x}}\big)^2}\Bigg]$
[Using quotient rule and chain rule]
$=\frac{(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}})\Big[\text{e}^{2\text{x}}\frac{\text{d}}{\text{dx}}(2\text{x})+\text{e}^{-2\text{x}}\frac{\text{d}}{\text{dx}}(-2\text{x})\Big]-\big(\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}\big)\Big[\text{e}^{2\text{x}}\frac{\text{d}}{\text{dx}}(2\text{x})-\text{e}^{-2\text{x}}\frac{\text{d}}{\text{dx}}(-2\text{x})\Big]}{\big(\text{e}^{2\text{x}}-\text{e}^{2\text{x}}\big)^2}$
$=\frac{\big(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}\big)\big(2\text{e}^{2\text{x}}-2\text{e}^{-2\text{e}}\big)-\big(\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}\big)\big(2\text{e}^{2\text{e}}+2\text{e}^{-2\text{x}}\big)}{\big(\text{e}^{2\text{e}}-\text{e}^{-2\text{x}}\big)^2}$
$=\frac{2\big(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}\big)^2-2\big(\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}\big)^2}{\big(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}\big)^2}$
$=\frac{2\big[\text{e}^{4\text{x}}+\text{e}^{-4\text{x}}-2\text{e}^{2\text{x}}\text{e}^{-2\text{x}}-\text{e}^{4\text{x}}-\text{e}^{-4\text{x}}-2\text{e}^{2\text{x}}\text{e}^{-2\text{x}}\big]}{\big(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}\big)^2}$
$=\frac{-8}{\big(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}\big)^2}$
So,
$\frac{\text{d}}{\text{dx}}\Big(\frac{\text{e}^{2\text{x}}+\text{e}^{-2\text{x}}}{\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}}\Big)=\frac{-8}{\big(\text{e}^{2\text{x}}-\text{e}^{-2\text{x}}\big)^2}$

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

Two cards are drawn simultaneously from a pack of 52 cards. Compute the mean and standard deviation of the number of kings.
→
Find the coordinate of the point P where the line through $\text{A(3, – 4, –5) and B (2, –3, 1)}$ crosses the plane passing through three points L(2, $\text{2, 1), M(3, 0, 1) and N(4, –1, 0).}$ Also, find the ratio in which P divides the line segment AB.
 
→
Find the equation of the plane through $(2, 3, -4)$ and $(1, -1, 3)$ and parallel to x-axis.
→
Solve the following systems of linear equations by cramer's rule:
2x - y = 17,
3x + 5y = 6
→
Prove that:
$\begin{vmatrix}\text{a}^2&2\text{ab}&\text{b}^2\\\text{b}^2&\text{a}^2&2\text{ab}\\2\text{ab}&\text{b}^2&\text{a}^2\end{vmatrix}=(\text{a}^3+\text{b}^3)^2$
→
Solve the following differential equation:$+ y = \cos x - \sin x.$
→
A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of 17.50 per package on nuts and 7 per package of bolts. How many packages of each should be produced each day so as to maximise his profits if he operates his machines for at the most 12 hours a day? Form the above as a linear programming problem and solve it graphically.
→
Find graphically, the maximum value of Z = 2x + 5y, subject to constraints given below:
$2\text{x}+4\text{y}\leq8$
$3\text{x}+\text{y}\leq6$
$\text{x}+\text{y}\leq4$
$\text{x}\geq0,\text{y}\geq0$
→
If $\text{x}=\sin^{-1}\Big(\frac{2\text{t}}{1+\text{t}^2}\Big)$ and $\text{y}=\tan^{-1}\Big(\frac{2\text{t}}{1-\text{t}^2}\Big),-1<\text{t}<1,$ prove that $\frac{\text{dy}}{\text{dx}}=1$
→
If $\text{y}=\log\frac{\text{x}^2+\text{x}+1}{\text{x}^2-\text{x}+1}+\frac{2}{\sqrt{3}}\tan^{-1}\Big(\frac{\sqrt{3}\text{x}}{1-\text{x}^2}\Big),$ find $\frac{\text{dy}}{\text{dx}}$
→