NCERT Solutions
Class 12 - Mathematics Part-1 - Chapter 5: CONTINUITY AND DIFFERENTIABILITYExercise 5.2
Class 12 - Mathematics Part-1 - Chapter 5: CONTINUITY AND DIFFERENTIABILITY
Question. 1
Differentiate with respect to \(x\): \(\sin(x^2+5)\).
Answer:
\(2x\cos(x^2+5)\)
Question. 2
Differentiate with respect to \(x\): \(\cos(\sin x)\).
Answer:
\(-\cos x\,\sin(\sin x)\)
Question. 3
Differentiate with respect to \(x\): \(\sin(ax+b)\).
Answer:
\(a\cos(ax+b)\)
Question. 4
Differentiate with respect to \(x\): \(\sec(\tan(\sqrt{x}))\).
Answer:
\(\dfrac{\sec(\tan\sqrt{x})\,\tan(\tan\sqrt{x})\,\sec^2\sqrt{x}}{2\sqrt{x}}\)
Question. 5
Differentiate with respect to \(x\): \(\dfrac{\sin(ax+b)}{\cos(cx+d)}\).
Answer:
\(a\cos(ax+b)\sec(cx+d)+c\sin(ax+b)\tan(cx+d)\sec(cx+d)\)
Question. 6
Differentiate with respect to \(x\): \(\cos(x^3)\cdot \sin^2(x^5)\).
Answer:
\(10x^4\sin(x^5)\cos(x^5)\cos(x^3)-3x^2\sin(x^3)\sin^2(x^5)\)
Question. 7
Differentiate with respect to \(x\): \(2\sqrt{\cot(x^2)}\).
Answer:
\(\dfrac{-2\sqrt{2}\,x}{\sin(x^2)\sqrt{\sin(2x^2)}}\)
Question. 8
Differentiate with respect to \(x\): \(\cos(\sqrt{x})\).
Answer:
\(\dfrac{-\sin\sqrt{x}}{2\sqrt{x}}\)
Question. 9
Prove that the function \(f\) given by \(f(x)=|x-1|\), \(x\in\mathbb{R}\), is not differentiable at \(x=1\).
Answer:
Question. 10
Prove that the greatest integer function defined by \(f(x)=[x]\), \(0<x<3\), is not differentiable at \(x=1\) and \(x=2\).