NCERT Solutions
Class 12 - Mathematics Part-1 - Chapter 5: CONTINUITY AND DIFFERENTIABILITYExercise 5.4
Class 12 - Mathematics Part-1 - Chapter 5: CONTINUITY AND DIFFERENTIABILITY
Question. 1
Differentiate \(\dfrac{e^x}{\sin x}\) with respect to \(x\).
Answer:
\(\dfrac{e^x(\sin x-\cos x)}{\sin^2 x}\), \(x \ne n\pi,\; n\in \mathbb{Z}\).
Question. 2
Differentiate \(e^{\sin^{-1}x}\) with respect to \(x\).
Answer:
\(\dfrac{e^{\sin^{-1}x}}{\sqrt{1-x^2}}\), \(x\in(-1,1)\).
Question. 3
Differentiate \(e^{x^3}\) with respect to \(x\).
Answer:
\(3x^2e^{x^3}\).
Question. 4
Differentiate \(\sin\big(\tan^{-1}(e^{-x})\big)\) with respect to \(x\).
Answer:
\(-\dfrac{e^{-x}\cos\big(\tan^{-1}(e^{-x})\big)}{1+e^{-2x}}\).
Question. 5
Differentiate \(\log(\cos e^x)\) with respect to \(x\).
Answer:
\(-e^x\tan(e^x)\), \(e^x\ne(2n+1)\dfrac{\pi}{2},\; n\in\mathbb{N}\).
Question. 6
Differentiate \(e^x + e^{x^2} + \cdots + e^{x^5}\) with respect to \(x\).
Answer:
\(e^x + 2xe^{x^2} + 3x^2e^{x^3} + 4x^3e^{x^4} + 5x^4e^{x^5}\).
Question. 7
Differentiate \(\sqrt{e^{\sqrt{x}}}\) with respect to \(x\), where \(x>0\).
Answer:
\(\dfrac{e^{\sqrt{x}}}{4\sqrt{x}e^{\sqrt{x}}}\), \(x>0\).
Question. 8
Differentiate \(\log(\log x)\) with respect to \(x\), where \(x>1\).
Answer:
\(\dfrac{1}{x\log x}\), \(x>1\).
Question. 9
Differentiate \(\dfrac{\cos x}{\log x}\) with respect to \(x\), where \(x>0\).
Answer:
\(-\dfrac{x\sin x\cdot\log x+\cos x}{x(\log x)^2}\), \(x>0\).
Question. 10
Differentiate \(\cos(\log x + e^x)\) with respect to \(x\), where \(x>0\).
Answer:
\(-\dfrac{1}{x}+e^x\sin(\log x+e^x)\), \(x>0\).