Continuity and Differentiability are fundamental concepts in calculus. Continuity ensures a function has no breaks or jumps, making it smooth. Differentiability means a function has a defined derivative, allowing for tangent calculations. A differentiable function is always continuous but not vice versa. These concepts are essential in solving calculus problems.
Class 12 Maths Chapter 5 Continuity and Differentiability is an important topic for the CBSE Exam 2024-25. It explains the concepts of continuity differentiability derivatives of inverse trigonometric functions and second-order derivatives. Understanding these concepts is essential for solving calculus problems and helps in building a strong foundation for higher mathematics and competitive exams.
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We are given y = tan⁻¹(e²ˣ) and need to find dy/dx.
Step 1: Differentiate both sides with respect to x
We differentiate the equation y = tan⁻¹(e²ˣ) using the chain rule. The derivative of tan⁻¹(u) with respect to u is 1/(1 + u²), so:
dy/dx = 1 / (1 + (e²ˣ)²) * d/dx(e²ˣ)
Step 2: Differentiate e²ˣ
The derivative of e²ˣ with respect to x is:
d/dx(e²ˣ) = 2e²ˣ
Step 3: Substitute into the derivative
Substitute this back into the expression for dy/dx:
dy/dx = 1 / (1 + e⁴ˣ) * 2e²ˣ
dy/dx = 2e²ˣ / (1 + e⁴ˣ)
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