If $x = \sqrt{2^{cosec^{-1} } t}$ and $y = \sqrt{2^{\sec^{-1}} t} (|t | \geq 1)$, then $\frac{dy}{dx}$ is equal to :
- $\frac{y}{x}$
- $\frac{x}{y}$
- $ - \frac{y}{x}$
- $ - \frac{x}{y}$
The Correct Option is C
Solution and Explanation
Now, $\frac{d y}{d x}=\frac{d y / d t}{d x / d t}$
$=\frac{\frac{1}{2 \sqrt{2 \sec ^{-1} t}} 2^{\sec ^{-1} t} \ln \left(\frac{1}{t \sqrt{t^{2}-1}}\right)}{-\frac{1}{2 \sqrt{2 cosec^{-1} t}} 2^{\text {cosec }^{-1}} \ln \left(\frac{1}{t \sqrt{t^{2}-1}}\right)} $
$=-\frac{\sqrt{2^{ ec ^{-1} t}}}{\sqrt{2^{\text {omee }^{-1} y}}}=\frac{-y}{x}$
Top Questions on Differentiability
Match the following:
In the following, \( [x] \) denotes the greatest integer less than or equal to \( x \).
Choose the correct answer from the options given below:- KCET - 2025
- Mathematics
- Differentiability
- The function f(x) is given by:
For x < 0:
f(x) = ex + axFor x ≥ 0:
f(x) = b(x - 1)2
The function is differentiable at x = 0. Then,- KCET - 2025
- Mathematics
- Differentiability
- If \[ y = \frac{\cos x}{1 + \sin x} \] then:
- KCET - 2025
- Mathematics
- Differentiability
- The number of 3-digit numbers, that are divisible by 2 and 3, but not divisible by 4 and 9, is.
- JEE Main - 2025
- Mathematics
- Differentiability
Let the function, \(f(x)\) = \(\begin{cases} -3ax^2 - 2, & x < 1 \\a^2 + bx, & x \geq 1 \end{cases}\) Be differentiable for all \( x \in \mathbb{R} \), where \( a > 1 \), \( b \in \mathbb{R} \). If the area of the region enclosed by \( y = f(x) \) and the line \( y = -20 \) is \( \alpha + \beta\sqrt{3} \), where \( \alpha, \beta \in \mathbb{Z} \), then the value of \( \alpha + \beta \) is:
- JEE Main - 2025
- Mathematics
- Differentiability
Questions Asked in JEE Main exam
- Atomic number of element with lowest first ionisation enthalpy is:
- JEE Main - 2025
- Periodic Table
- Let \( z_1, z_2, z_3 \) be three complex numbers on the circle \( |z| = 1 \) with \( \arg(z_1) = -\frac{\pi}{4}, \arg(z_2) = 0 \) and \( \arg(z_3) = \frac{\pi}{4} \). If \( |z_1 \overline{z_2} + z_2 \overline{z_3} + z_3 \overline{z_1}|^2 = \alpha + \beta \sqrt{2} \), where \( \alpha, \beta \in \mathbb{Z} \), then the value of \( \alpha^2 + \beta^2 \) is :
- JEE Main - 2025
- complex numbers
- The kinetic energy of translation of the molecules in 50 g of CO\(_2\) gas at 17°C is:
- JEE Main - 2025
- The Kinetic Theory of Gases
- Assume a living cell with 0.9%(\(w/w\)) of glucose solution (aqueous). This cell is immersed in another solution having equal mole fraction of glucose and water. (Consider the data up to first decimal place only) The cell will:
- JEE Main - 2025
- osmosis
- If the system of equations \[ x + 2y - 3z = 2 \] \[ 2x + \lambda y + 5z = 5 \] \[ 4x + 3y + \mu z = 33 \] has infinite solutions, then \( \lambda + \mu \) is equal to
- JEE Main - 2025
- Miscellaneous
Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.