let f x log x 1 x ne 1 the value of f 12 895021306

Let $$f\left( x \right) = \log \left| {x - 1} \right|,\,x \ne 1.$$ The value of $$f'\left( {\frac{1}{2}} \right) = ?$$

A. is $$ - 2$$

B. is $$2$$

C. does not exist

D. none of these

Answer : is $$ - 2$$

Solution :
$$\eqalign{ & {\text{RH derivative}} = \mathop {\lim }\limits_{h \to 0} \frac{{\log \left| {\frac{1}{2} + h - 1} \right| - \log \left| {\frac{1}{2} - 1} \right|}}{h} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{\log \left| {h - \frac{1}{2}} \right| - \log \frac{1}{2}}}{h} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{\log \left( {\frac{1}{2} - h} \right) + \log \,2}}{h} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{\frac{1}{2} - h}}\left( { - 1} \right) + 0}}{1} \cr & = - 2 \cr & {\text{LH derivative}} = \mathop {\lim }\limits_{h \to 0} \frac{{\log \left| {\frac{1}{2} - h - 1} \right| - \log \left| {\frac{1}{2} - 1} \right|}}{{ - h}} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{\log \left( {\frac{1}{2} + h} \right) - \log \frac{1}{2}}}{{ - h}} \cr & = \mathop {\lim }\limits_{h \to 0} \frac{{\frac{1}{{\frac{1}{2} + h}} - 0}}{{ - 1}} \cr & = - 2 \cr & {\bf{Alternate\,method :}} \cr & {\text{Around}}\,\,x = \frac{1}{2},\,f\left( x \right) = \log \left( {1 - x} \right) \cr & {\text{So, }}f'\left( x \right) = \frac{{ - 1}}{{1 - x}} \cr & \therefore f'\left( {\frac{1}{2}} \right) = \frac{{ - 1}}{{1 - \frac{1}{2}}} = - 2 \cr} $$

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 $$x$$

View Answer

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

View Answer

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

View Answer

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

View Answer

Let $$f\left( x \right) = \log \left| {x - 1} \right|,\,x \ne 1.$$ The value of $$f'\left( {\frac{1}{2}} \right) = ?$$

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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Let $$f\left( x \right) = \log \left| {x - 1} \right|,\,x \ne 1.$$ The value of $$f'\left( {\frac{1}{2}} \right) = ?$$

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-

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