if y a x a x b x x b a x x b 155027793

If $$y = \frac{{\left( {a - x} \right)\sqrt {a - x} - \left( {b - x} \right)\sqrt {x - b} }}{{\sqrt {a - x} + \sqrt {x - b} }},$$ then $$\frac{{dy}}{{dx}}$$ wherever it is defined is :

A. $$\frac{{x + \left( {a + b} \right)}}{{\sqrt {\left( {a - x} \right)\left( {x - b} \right)} }}$$

B. $$\frac{{2x - a - b}}{{2\sqrt {a - x} \sqrt {x - b} }}$$

C. $$ - \frac{{\left( {a + b} \right)}}{{2\sqrt {\left( {a - x} \right)\left( {x - b} \right)} }}$$

D. $$\frac{{2x + \left( {a + b} \right)}}{{2\sqrt {\left( {a - x} \right)\left( {x - b} \right)} }}$$

Answer : $$\frac{{2x - a - b}}{{2\sqrt {a - x} \sqrt {x - b} }}$$

Solution :
$$\eqalign{ & y = \frac{{{{\left( {a - x} \right)}^{\frac{3}{2}}} + {{\left( {x - b} \right)}^{\frac{3}{2}}}}}{{\sqrt {a - x} + \sqrt {x - b} }} \cr & = \frac{{\left( {\sqrt {a - x} + \sqrt {x - b} } \right)\left( {a - x - \sqrt {a - x} \sqrt {x - b} + x - b} \right)}}{{\sqrt {a - x} + \sqrt {x - b} }} \cr & = a - b - \sqrt {a - x} \sqrt {x - b} \cr & {\text{or }}\frac{{dy}}{{dx}} = \frac{1}{{2\sqrt {a - x} }}\sqrt {x - b} - \frac{1}{{2\sqrt {x - b} }}\sqrt {a - x} \cr & = \frac{{2x - a - b}}{{2\sqrt {a - x} \sqrt {x - b} }} \cr} $$

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

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

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

If $$y = \frac{{\left( {a - x} \right)\sqrt {a - x} - \left( {b - x} \right)\sqrt {x - b} }}{{\sqrt {a - x} + \sqrt {x - b} }},$$ then $$\frac{{dy}}{{dx}}$$ wherever it is defined 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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Differentiability and Differentiation

Practice More MCQ Question on Maths Section

If $$y = \frac{{\left( {a - x} \right)\sqrt {a - x} - \left( {b - x} \right)\sqrt {x - b} }}{{\sqrt {a - x} + \sqrt {x - b} }},$$ then $$\frac{{dy}}{{dx}}$$ wherever it is defined 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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Releted MCQ Question on
Calculus >> Differentiability and Differentiation

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Differentiability and Differentiation