NCERT Class 12 Maths
Chapter 5: Continuity And Differentiability
22 Questions Solved
NCERT Solutions for Class 12 Maths Chapter 5 Continuity And Differentiability — Miscellaneous Exercise
Click any question below for its step-by-step NCERT textbook solution, concept hints, and worked explanation.
Question 1
Differentiate \(\left(3 x^{2} - 9 x + 5\right)^{9}\) with respect to x; on R.
Question 2
Differentiate \(\sin^{3}{\left(x \right)} + \cos^{6}{\left(x \right)}\) with respect to x; on R.
Question 3
Differentiate \(\left(5 x\right)^{3 \cos{\left(2 x \right)}}\) with respect to x; for x>0.
Question 4
Differentiate \(\operatorname{asin}{\left(x^{\frac{3}{2}} \right)}\) with respect to x; for 0<x<1; at 0 the right derivative is 0…
Question 5
Differentiate \(\frac{\operatorname{acos}{\left(\frac{x}{2} \right)}}{\sqrt{2 x + 7}}\) with respect to x; for −2<x<2.
Question 6
Differentiate cot⁻¹[(√(1+sin x)+√(1−sin x))/(√(1+sin x)−√(1−sin x))], 0<x<π/2.
Question 7
Differentiate \(\log{\left(x \right)}^{\log{\left(x \right)}}\) with respect to x; for x>1.
Question 8
Differentiate \(\cos{\left(a \cos{\left(x \right)} + b \sin{\left(x \right)} \right)}\) with respect to x; on R.
Question 9
Differentiate \(\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{\sin{\left(x \right)} - \cos{\left(x \right)}}\) with…
Question 10
Differentiate \(a^{a} + a^{x} + x^{a} + x^{x}\) with respect to x; for x>0 and fixed a>0.
Question 11
Differentiate \(x^{x^{2} - 3} + \left(x - 3\right)^{x^{2}}\) with respect to x; for x>3.
Question 12
Find dy/dx for y=12(1−cos t), x=10(t−sin t), −π/2<t<π/2.
Question 13
Differentiate y=sin⁻¹x+sin⁻¹√(1−x²), 0<x<1.
Question 14
If x√(1+y)+y√(1+x)=0, −1<x<1, prove y′=−1/(1+x)².
Question 15
If (x−a)²+(y−b)²=c², c>0, examine [1+(y′)²]^(3/2)/y″ and its independence of a,b.
Question 16
If cos y=x cos(a+y), cos a≠±1, prove y′=cos²(a+y)/sin a.
Question 17
For x=a(cos t+t sin t), y=a(sin t−t cos t), find d²y/dx².
Question 18
For f(x)=|x|³, show f″ exists for every real x and find it.
Question 19
Differentiate sin(A+B)=sin A cos B+cos A sin B to obtain the cosine sum formula.
Question 20
Does an everywhere-continuous function exist which is not differentiable at exactly two points? Justify.
Question 21
If y=det [[f(x),g(x),h(x)],[l,m,n],[a,b,c]], with l,m,n,a,b,c constants, prove y′=det [[f′(x),g′(x),h′(x)],[l,m,n],[a,b,c]].
Question 22
For y=e^(a cos⁻¹x), −1≤x≤1, show (1−x²)y″−xy′−a²y=0.