Differentiate the following w.r.t. x: log(cos ex)
Solution and Explanation
Let y=log(cos ex)
By using the chain rule, we obtain
\(\frac{dy}{dx}=\frac{d}{dx}[log(cos e^x]\)
=\(\frac{1}{cose^x}.\frac{d}{dx}(cose^x)\)
=\(\frac{1}{cose^x}.(-sin e^x).\frac{d}{dx}(e^x)\)
=\(\frac{-sin e^x}{cos e^x}.e^x\)
=-\(e^x tan e^x,e^x\)≠\((2n+1)\frac{\pi}{2},\)\(nεN\)
Top Questions on Continuity and differentiability
- If \[ f(x) = \begin{cases} \frac{\log(1 + ax) + \log(1 - bx)}{x}, & \text{for } x \neq 0 \\ k, & \text{for } x = 0 \end{cases} \] is continuous at $x = 0$, then the value of $k$ is:
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Match List-I with List-II
List-I List-II (A) \( f(x) = |x| \) (I) Not differentiable at \( x = -2 \) only (B) \( f(x) = |x + 2| \) (II) Not differentiable at \( x = 0 \) only (C) \( f(x) = |x^2 - 4| \) (III) Not differentiable at \( x = 2 \) only (D) \( f(x) = |x - 2| \) (IV) Not differentiable at \( x = 2, -2 \) only
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- Let \( y=\sin(\cos(x^2)) \). Find \( \frac{dy}{dx} \) at \( x=\frac{\sqrt{\pi}}{2} \).
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Questions Asked in CBSE CLASS XII exam
Reactant ‘A’ underwent a decomposition reaction. The concentration of ‘A’ was measured periodically and recorded in the table given below:
Based on the above data, predict the order of the reaction and write the expression for the rate law.- CBSE CLASS XII - 2025
- Chemical Kinetics
- Atharv and Anmol were partners in a firm sharing profits and losses in the ratio of 5 : 2. Their Balance Sheet as at 31st March, 2024 was as follows:
Balance Sheet of Atharv and Anmol as at 31st March, 2024
Surya was admitted for \(\frac{2}{7}\) share in profits. Terms:Liabilities Amount (₹) Assets Amount (₹) Capitals: Fixed Assets 14,00,000 Atharv 8,00,000 Stock 4,90,000 Anmol 4,00,000 Debtors 5,60,000 General Reserve 3,50,000 Cash 10,000 Creditors 9,10,000 Total 24,60,000 Total 24,60,000
New ratio: Atharv, Anmol, Surya = 4 : 1 : 2
Fixed Assets decreased 10%
Stock sold ₹ 4,20,000
Surya brings ₹ 3,00,000 capital, ₹ 2,00,000 goodwill
Capital adjusted to Surya's capital Prepare Revaluation A/c and Capital A/cs.- CBSE CLASS XII - 2025
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- In Bohr’s model of hydrogen atom, find the percentage change in the radius of its orbit when an electron makes a transition from \( n = 3 \) state to \( n = 2 \) state.
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- A 1 cm segment of a wire lying along the x-axis carries a current of 0.5 A along the \( +x \)-direction. A magnetic field \( \vec{B} = (0.4 \, \text{mT} \hat{j}) + (0.6 \, \text{mT} \hat{k}) \) is switched on. The force acting on the segment is:
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- (a) Consider the so-called ‘D-T reaction’ (Deuterium-Tritium reaction).
In a thermonuclear fusion reactor, the following nuclear reaction occurs: \[ \ ^{2}_1 \text{H} + \ ^{3}_1 \text{H} \longrightarrow \ ^{4}_2 \text{He} + \ ^{1}_0 \text{n} + Q \] Find the amount of energy released in the reaction.
% Given data Given:
\( m\left(^{2}_1 \text{H}\right) = 2.014102 \, \text{u} \)
\( m\left(^{3}_1 \text{H}\right) = 3.016049 \, \text{u} \)
\( m\left(^{4}_2 \text{He}\right) = 4.002603 \, \text{u} \)
\( m\left(^{1}_0 \text{n}\right) = 1.008665 \, \text{u} \)
\( 1 \, \text{u} = 931 \, \text{MeV}/c^2 \)- CBSE CLASS XII - 2025
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.