If u = f( x^3 ),v = g( x^2 ),f'( x ) = cos x and g'( x ) = sin x then dudv is :

dudv = dudxdvdx = f( x^3 ).3x^2g( x^2 ).2x = cos x^3.3x^2sin x^2.2x = 32xcos x^3.cosecx^2

if u f x pow 3 v f x pow 2 f x cos x and g x sin x then du

If $$u = f\left( {{x^3}} \right),\,v = g\left( {{x^2}} \right),\,f'\left( x \right) = \cos \,x$$ and $$g'\left( x \right) = \sin \,x$$ then $$\frac{{du}}{{dv}}$$ is :

A. $$\frac{3}{2}x\,\cos \,{x^3}.\,{\text{cosec}}\,{x^2}$$

B. $$\frac{2}{3}\sin \,{x^3}.\sec \,{x^2}$$

C. $$\tan \,x$$

D. none of these

Answer : $$\frac{3}{2}x\,\cos \,{x^3}.\,{\text{cosec}}\,{x^2}$$

Solution :
$$\frac{{du}}{{dv}} = \frac{{\frac{{du}}{{dx}}}}{{\frac{{dv}}{{dx}}}} = \frac{{f'\left( {{x^3}} \right).3{x^2}}}{{g'\left( {{x^2}} \right).2x}} = \frac{{\cos \,{x^3}.3{x^2}}}{{\sin \,{x^2}.2x}} = \frac{3}{2}x\,\cos \,{x^3}.{\text{cosec}}\,{x^2}$$

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

If $$u = f\left( {{x^3}} \right),\,v = g\left( {{x^2}} \right),\,f'\left( x \right) = \cos \,x$$ and $$g'\left( x \right) = \sin \,x$$ then $$\frac{{du}}{{dv}}$$ 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:

Practice More Releted MCQ Question on
Differentiability and Differentiation

Practice More MCQ Question on Maths Section

If $$u = f\left( {{x^3}} \right),\,v = g\left( {{x^2}} \right),\,f'\left( x \right) = \cos \,x$$ and $$g'\left( x \right) = \sin \,x$$ then $$\frac{{du}}{{dv}}$$ 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:

Practice More Releted MCQ Question on Differentiability and Differentiation

Practice More MCQ Question on Maths Section

Releted MCQ Question on
Calculus >> Differentiability and Differentiation

Practice More Releted MCQ Question on
Differentiability and Differentiation