the derivative of ln x sin x with respect to x cos x is

The derivative of $${\text{ln}}\left( {x + \sin \,x} \right)$$ with respect to $$\left( {x + \cos \,x} \right)$$ is :

A. $$\frac{{1 + \cos \,x}}{{\left( {x + \sin \,x} \right)\left( {1 - \sin \,x} \right)}}$$

B. $$\frac{{1 - \cos \,x}}{{\left( {x + \sin \,x} \right)\left( {1 + \sin \,x} \right)}}$$

C. $$\frac{{1 - \cos \,x}}{{\left( {x - \sin \,x} \right)\left( {1 + \cos \,x} \right)}}$$

D. $$\frac{{1 + \cos \,x}}{{\left( {x - \sin \,x} \right)\left( {1 - \cos \,x} \right)}}$$

Answer : $$\frac{{1 + \cos \,x}}{{\left( {x + \sin \,x} \right)\left( {1 - \sin \,x} \right)}}$$

Solution :
$$\eqalign{ & {\text{ln}}{\left( {x + \sin \,x} \right)^1} = y\,\,\,\,\,\,\,\,\,\left( {{\text{say}}} \right) \cr & \frac{{dy}}{{dx}} = \frac{1}{{\left( {x + \sin \,x} \right)}}\left( {1 + \cos \,x} \right) = \frac{{\left( {1 + \cos \,x} \right)}}{{\left( {x + \sin \,x} \right)}} \cr & x + \cos \,x = z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {{\text{say}}} \right) \cr & \frac{{dz}}{{dx}} = \left( {1 - \sin \,x} \right) \cr & {\text{derivative of ln}}\left( {x + \sin \,x} \right){\text{ w}}{\text{.r}}{\text{.t }}\left( {x + \cos \,x} \right){\text{ is}} \cr & \frac{{dy}}{{dz}} = \frac{{\left( {1 + \cos \,x} \right)}}{{\left( {x + \sin \,x} \right)\left( {1 - \sin \,x} \right)}} \cr} $$

Releted MCQ Question on
Calculus >> Differentiability and Differentiation

Releted Question 1

There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-

A. $$f''\left( x \right) > 0$$ for all $$x$$

B. $$ - 1 < f''\left( x \right) < 0$$ for all $$x$$

C. $$ - 2 \leqslant f''\left( x \right) \leqslant - 1$$ for all $$x$$

D. $$f''\left( x \right) < - 2$$ for all $$x$$

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Releted Question 2

If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-

A. $$-5$$

B. $$\frac{1}{5}$$

C. $$5$$

D. none of these

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Releted Question 3

Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-

A. $$8f'\left( 1 \right)$$

B. $$4f'\left( 1 \right)$$

C. $$2f'\left( 1 \right)$$

D. $$f'\left( 1 \right)$$

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Releted Question 4

Let [.] denote the greatest integer function and $$f\left( x \right) = \left[ {{{\tan }^2}x} \right],$$ then:

A. $$\mathop {\lim }\limits_{x \to 0} f\left( x \right)$$ does not exist

B. $$f\left( x \right)$$ is continuous at $$x = 0$$

C. $$f\left( x \right)$$ is not differentiable at $$x =0$$

D. $$f'\left( 0 \right) = 1$$

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The derivative of $${\text{ln}}\left( {x + \sin \,x} \right)$$ with respect to $$\left( {x + \cos \,x} \right)$$ is :

Releted MCQ Question on
Calculus >> Differentiability and Differentiation

There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-

If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-

Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-

Let [.] denote the greatest integer function and $$f\left( x \right) = \left[ {{{\tan }^2}x} \right],$$ then:

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The derivative of $${\text{ln}}\left( {x + \sin \,x} \right)$$ with respect to $$\left( {x + \cos \,x} \right)$$ is :

Releted MCQ Question on Calculus >> Differentiability and Differentiation

There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-

If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-

Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-

Let [.] denote the greatest integer function and $$f\left( x \right) = \left[ {{{\tan }^2}x} \right],$$ then:

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