Sum and difference rule Quiz 1

Question 1

Solve for $$\dfrac{d}{dx}\left( \sin \left( 2{{\coth }^{-1}}x \right) \right)$$.

Accepted Answer

$$\cos \left( 2{{\coth }^{-1}}x \right)\cdot \left( \dfrac{2}{1-{{x}^{2}}} \right)$$

Answer

$$\cos \left( 2{{\coth }^{-1}}x \right)\cdot \left( \dfrac{-2}{1-{{x}^{2}}} \right)$$

Answer

$$\cos \left( 2{{\coth }^{-1}}x \right)\cdot \left( \dfrac{2}{\sqrt{1-{{x}^{2}}}} \right)$$

Answer

$$\cos \left( 2{{\coth }^{-1}}x \right)\cdot \left( \dfrac{-2}{\sqrt{1-{{x}^{2}}}} \right)$$

Question 2

Find differentiation of $$\dfrac{d}{dx}\left( \coth x+2{{\sinh }^{-1}}\left( x \right)+\dfrac{\sqrt{57}}{2456} \right)$$.

Accepted Answer

$$-	ext{cosec}{{	ext{h}}^{2}}\left( x ight)+2\dfrac{1}{\sqrt{\left( {{x}^{2}}+1 ight)}}$$

Answer

$$	ext{cosec}{{	ext{h}}^{2}}\left( x ight)-2\dfrac{1}{\sqrt{\left( {{x}^{2}}+1 ight)}}$$

Answer

$$-	ext{sec}{{	ext{h}}^{2}}\left( x ight)-2\dfrac{1}{\sqrt{\left( {{x}^{2}}+1 ight)}}$$

Answer

$$-	ext{cosec}{{	ext{h}}^{2}}\left( x ight)+2\dfrac{1}{\sqrt{\left( {{x}^{2}}-1 ight)}}$$

Question 3

Use differentiation techniques to solve $$\dfrac{d}{dx}\left( \sin \left( \ln \left( {{\sec }^{-1}}x+	anh x ight) ight) ight)$$.

Accepted Answer

$$\cos \left(\ln \left(\sec^{-1} \left(xight)+	anh \left(xight)ight)ight)\dfrac{1+\sec h ^2\left(xight)\sqrt{x^2}\sqrt{x^2-1}}{\sqrt{x^2}\left(\sec^{-1} \left(xight)+	anh \left(xight)ight)\sqrt{x^2-1}}$$

Answer

$$\cos \left(\ln \left(-\sec^{-1} \left(xight)+	anh \left(xight)ight)ight)\dfrac{1+\sec h ^2\left(xight)\sqrt{x^2}\sqrt{x^2-1}}{\sqrt{x^2}\left(\sec^{-1} \left(xight)+	anh \left(xight)ight)\sqrt{x^2-1}}$$

Answer

$$\cos \left(\ln \left(\sec^{-1} \left(xight)+	anh \left(xight)ight)ight)\dfrac{1+\sec h ^2\left(xight)\sqrt{x^2}\sqrt{x^2-1}}{\sqrt{x^2}\left(\sec^{-1} \left(xight)-	anh \left(xight)ight)\sqrt{x^2-1}}$$

Answer

$$\cos \left(\ln \left(-\sec^{-1} \left(xight)+	anh \left(xight)ight)ight)\dfrac{1+\sec h ^2\left(xight)\sqrt{x^2}\sqrt{x^2-1}}{\sqrt{x^2}\left(\sec^{-1} \left(xight)-	anh \left(xight)ight)\sqrt{x^2-1}}$$

Question 4

What will be the differentiation of $$\dfrac{d}{dx}\left( {{\cot }^{-1}}\left( \ln x \right) \right)$$.

Accepted Answer

$$\dfrac{-1}{x\left( 1+{{\left( \ln x \right)}^{2}} \right)}$$

Answer

$$\dfrac{1}{x\left( 1-{{\left( \ln x \right)}^{2}} \right)}$$

Answer

$$\dfrac{-1}{x\left( 1-{{\left( \ln x \right)}^{2}} \right)}$$

Answer

$$\dfrac{1}{x\left( 1+{{\left( \ln x \right)}^{2}} \right)}$$

Question 5

What will be the answer of this term $$\dfrac{d}{dx}\left( {{\sec }^{-1}}\left( 	ext{cosech}\left( x ight) ight) ight)$$.

Accepted Answer

$$\dfrac{1}{\left| 	ext{cosech}\left( x ight) ight|\sqrt{{{\left( 	ext{cosech}\left( x ight) ight)}^{2}}-1}}\cdot \left( 	ext{cosech}x\coth x ight)$$

Answer

$$\dfrac{1}{\left| {{x}^{	ext{cosech}\left( x ight)}} ight|\sqrt{{{\left( {{x}^{	ext{cosech}\left( x ight)}} ight)}^{2}}+1}}\cdot \left( {{x}^{	ext{cosech}\left( x ight)}}\cdot \left[ \dfrac{1}{x}+\ln \left( x ight)\cdot \left( 	ext{sech }x	an x ight)+	ext{sech}\left( x ight)\cdot \dfrac{1}{x} ight] ight)$$

Answer

$$\dfrac{1}{\sqrt{{{\left( {{x}^{	ext{cosech}\left( x ight)}} ight)}^{2}}+1}}\cdot \left( {{x}^{	ext{-cosech}\left( x ight)}}\cdot \left[ \dfrac{1}{x}+	ext{cosech}\left( x ight)\cdot \dfrac{1}{x} ight] ight)$$

Answer

$$\dfrac{1}{\left| {{x}^{	ext{cosech}\left( x ight)}} ight|\sqrt{{{\left( {{x}^{	ext{cosech}\left( x ight)}} ight)}^{2}}-1}}\cdot \left( {{x}^{	ext{cosech}\left( x ight)}}\cdot \left[ \dfrac{1}{x}+\ln \left( x ight)\cdot \left( -	ext{cosech }x\coth x ight)+	ext{coth}\left( x ight)\cdot \dfrac{1}{x} ight] ight)$$