Download App

CONTINUITY AND DIFFERENTIABILITY — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsCONTINUITY AND DIFFERENTIABILITY4 Marks
Question
If $\text{x}=\Big(\text{t}+\frac{1}{\text{t}}\Big)^\text{a},\text{y}=\text{a}^{\text{t}+\frac{1}{\text{t}}},$ find $\frac{\text{dy}}{\text{dx}}$
✓

Answer

Here, $\text{x}=\Big(\text{t}+\frac{1}{\text{t}}\Big)^{\text{a}}$
Differentiating it with respect to t using chain rule,
$\frac{\text{dx}}{\text{dt}}=\frac{\text{d}}{\text{dt}}\left[\Big(\text{t}+\frac{1}{\text{t}}\Big)^{\text{a}}\right]$
$=\text{a}\Big(\text{t}+\frac{1}{\text{t}}\Big)^{\text{a-1}}\frac{\text{d}}{\text{dt}}\Big(\text{t}+\frac{1}{\text{t}}\Big)$
$\frac{\text{dx}}{\text{dt}}=\text{a}\Big(\text{t}+\frac{1}{\text{t}}\Big)^{\text{a-1}}\Big(\text{1}-\frac{1}{\text{t}^{2}}\Big)\ .....(\text{i})$
And, $\text{y}=\text{a}^{(\text{t}+\frac{1}{\text{t}})}$
Differentiating it with respect to t using chain rule,
$\frac{\text{dy}}{\text{dt}}=\frac{\text{d}}{\text{dt}}\bigg[\text{a}^{\big(\text{t}+\frac{1}{\text{t}}\big)}\bigg]$
$=\frac{\text{d}}{\text{dt}}\text{a}^{\big(\text{t}+\frac{1}{\text{t}}\big)}\times\log\frac{\text{d}}{\text{dt}}\left(\text{t}+\frac{1}{\text{t}}\right)$
$\frac{\text{dy}}{\text{dt}}=\text{a}^{\big(\text{t}+\frac{1}{\text{t}}\big)}\times\log\text{a}\Big(\text{t}-\frac{1}{\text{t}^{2}}\Big)\ .....(\text{ii})$
Dividing equation (ii) by (i),
$\frac{\frac{\text{dy}}{\text{dt}}}{\frac{\text{dx}}{\text{dt}}}=\frac{\text{a}^{\big(\text{t}+\frac{1}{\text{t}}\big)}\times\log\text{a}\left(1-\frac{1}{\text{t}^{2}}\right)}{\text{a}\big(\text{t}+\frac{1}{\text{t}}\big)^{\text{a}-1}\big(1-\frac{1}{\text{t}}^{2}\big)}$
$\frac{\text{dy}}{\text{dx}}=\frac{\text{a}^{\big(\text{t}+\frac{1}{\text{t}}\big)}\times\log\text{a}}{\text{a}\big(\text{t}+\frac{1}{\text{t}}\big)^{\text{a}-1}}$

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 "4 Marks" 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

Find the maximum and the minimum values, if any, without using derivaives of the following functions:

f(x) = -|x + 1| + 3 on R.

→
By examining the chest X-ray, probability that T.B. is detected when a person is actually suffering is 0.99. The probability that the doctor diagnoses incorrectly that a person has T.B. on the basic of X-ray is 0.001. In a certain city 1 in 1000 persons suffers from T.B. A person is selected at random is diagnosed to have T.B. what is the chance that he actually has T.B.?
→
Let R be the relation over the set of all straight lines in a plane such that  $\text{l}_1\text{Rl}_2\Leftrightarrow\text{l}_1\bot\text{l}_2.$ Then, R is:
  1. Symmetric.
  2. Reflexive.
  3. Transitive.
  4. An equivalence relation.
→
If $\text{y}=\text{cosec}^{-1}\text{x},\text{x}>1$ prove that $\text{x}(\text{x}^2-1)\frac{\text{d}^2\text{y}}{\text{dx}^2}+(2\text{x}^2-1)\frac{\text{dy}}{\text{dx}}=0.$
→
Evaluate the following intregals:
$\int\frac{\text{x}^2}{\text{x}^4-\text{x}^2-12}\ \text{dx}$
→
Find the equation of the plane that is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z - 4 = 0, 2x + y - z + 5 = 0.
→
Solve the following differential equation:
$\text{y}^2\frac{\text{dx}}{\text{dy}}+\text{x}-\frac{1}{\text{y}}=0$
→
If $\text{A}=\begin{bmatrix}0&-\text{x}\\\text{x}&0\end{bmatrix},\ \text{B}=\begin{bmatrix}0&1\\1&0\end{bmatrix}$ and x2 = -l, then show that (A + B)2 = A2 + B2.
→
A bag A contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B. If two balls are drawn at random (without replacement) from the selected bag, find the probability of one of them being red and another black.
→
The probability that a student selected at random from a class will pass in Mathematics is $\frac{4}{5}$, and the probability that he/ she passes in Mathematics and Computer Science is $\frac{1}{2}$. What is the probability that he/ she will pass in Computer Science if it is known that he/ she has passed in Mathematics?
→