Higher Order Derivatives
Differentiation
2
JEE Qs
8%
Hard
60
min
Master systematic differentiation techniques and pattern recognition for n-th derivatives, paying close attention to parametric and implicit differentiation intricacies.
🧮 Key Formulas
✅ Key Points for JEE
- 1Familiarize yourself with different notations for higher-order derivatives: d²y/dx², y'', y₂, D²y, etc.
- 2For second derivatives of parametric functions, apply the chain rule carefully: d²y/dx² = (d/dt (dy/dx)) / (dx/dt), which is NOT (d²y/dt²) / (d²x/dt²).
- 3When proving relations involving higher-order derivatives, it is often useful to substitute lower-order derivative expressions and simplify strategically.
- 4Leibniz's Theorem provides a systematic way to find the n-th derivative of a product of two functions, especially when n is large.
- 5For finding the n-th derivatives of common functions (e.g., e^ax, sin(ax+b), cos(ax+b), (ax+b)^m, 1/(ax+b)), look for patterns after calculating the first few derivatives.
⚠️ Common Mistakes
- ✕Incorrectly applying the chain rule or product/quotient rule repeatedly, leading to errors in algebraic simplification.
- ✕Using the incorrect formula for the second derivative of parametric functions: d²y/dx² ≠ (d²y/dt²) / (d²x/dt²).
- ✕Failing to recognize when implicit differentiation is required or applying it partially for implicit functions.
📝 Practice Questions
See allQ71.Let f(x) = x5 + 2x3 + 3x + 1, x ∈R , and g(x) be a function such that g(f(x)) = x for all x ∈R . Then g(7) g′(7) is equal to : (1) 14 (2) 42 (3) 7 (4) 1
Q73.If f(x) = {x30 sin, x (= 0 (1) f ′′ ( π2 ) = 24−π22π (2) f ′′ ( π2 ) = 12−π22π (3) f ′′(0) = 1 (4) f ′′(0) = 0
Q73.Let 𝑓𝑥= 2𝑥2 + 5𝑥- 3, 𝑥∈𝑅. If 𝑚 and 𝑛 denote the number of points where 𝑓 is not continuous and not differentiable respectively, then 𝑚+ 𝑛 is equal to: (1) 5 (2) 2 (3) 0 (4) 3
Q74.Let f(x) = x5 + 2ex/4 for all x ∈R. Consider a function g(x) such that (g ∘f)(x) = x for all x ∈R. Then the value of 8g′(2) is : (1) 2 (2) 8 (3) 4 (4) 16 is equal to :
Q74.Let β(m, n) = ∫10 xm−1(1 −x)n−1 dx, m, n > 0 . If ∫10 (1 −x10) dx = a × β(b, c), then 100(a + b + c) equals____ (1) 1021 (2) 2120 (3) 2012 (4) 1120 JEE Main 2024 (05 Apr Shift 2) JEE Main Previous Year Paper
Q72.Suppose for a differentiable function h, h(0) = 0, h(1) = 1 and h′(0) = h′(1) = 2. If g(x) = h (ex)eh(x) , then g′(0) is equal to: (1) 5 (2) 4 (3) 8 (4) 3
NCERT Chapters
- Class 12 Maths Ch 5: Continuity and Differentiability