Product rule Quiz 1

Question 1

Find $$\dfrac{dz}{dy}$$, if $$z{{y}^{3}}=\log y$$.

Accepted Answer

$${{y}^{-4}}\left( 1-3\log y \right)$$

Answer

$${{y}^{4}}\left( 1-3\log y \right)$$

Answer

$${{y}^{-4}}\left( 1-4\log y \right)$$

Answer

$${{y}^{2}}-3\dfrac{\log y}{{{y}^{3}}}$$

Question 2

Which of the following options is true if $$f(x)=\left( 7{{x}^{7}}+4{{x}^{2}}+1 \right)\left( 2{{x}^{4}}+3{{x}^{2}} \right)?$$

Accepted Answer

$$f'(x)=\left( 7{{x}^{7}}+4{{x}^{2}}+1 \right)\left( 8{{x}^{3}}+6x \right)+\left( 2{{x}^{4}}+3{{x}^{2}} \right)\left( 49{{x}^{6}}+8x \right)$$

Answer

$$f'(x)=\left( 7{{x}^{7}}+4{{x}^{2}}+1 \right)\left( 8{{x}^{4}}+6x \right)+\left( 2{{x}^{4}}+3{{x}^{2}} \right)\left( 49{{x}^{6}}+8x \right)$$

Answer

$$f'(x)=\left( 49{{x}^{6}}+4{{x}^{2}}+1 \right)\left( 8{{x}^{3}}+6x \right)+\left( 2{{x}^{4}}+3{{x}^{2}} \right)\left( 49{{x}^{6}}+8x \right)$$

Answer

$$f'(x)=\left( 7{{x}^{7}}+4{{x}^{2}}+1 \right)\left( 8{{x}^{3}}+6x \right)+\left( 2{{x}^{4}}+3{{x}^{2}} \right)\left( 49{{x}^{6}}+8 \right)$$

Question 3

Find $$y'$$, if $$y=\sqrt{x}\log x$$.

Accepted Answer

$$\dfrac{1}{2\sqrt{x}}\left( 2+\log x \right)$$

Answer

$$\dfrac{1}{\sqrt{x}}\left( 2+\log x \right)$$

Answer

$$\dfrac{1}{2\sqrt{x}}\left( 1+\log x \right)$$

Answer

$$\dfrac{1}{2\sqrt{x}}\left( 1+2\log x \right)$$

Question 4

Find $$y'$$, if $$y={{x}^{-2}}\left( 7+4{{x}^{-3}} \right)$$.

Accepted Answer

$$-\dfrac{14}{{{x}^{3}}}-\dfrac{20}{{{x}^{6}}}$$

Answer

$$-\dfrac{14}{{{x}^{3}}}+\dfrac{20}{{{x}^{6}}}$$

Answer

$$-\dfrac{14}{{{x}^{4}}}-\dfrac{20}{{{x}^{6}}}$$

Answer

$$-\dfrac{14}{{{x}^{3}}}-\dfrac{25}{{{x}^{6}}}$$

Question 5

Find $$y'$$, if $$y={{e}^{xz}}z\cos x$$ where $$z=\sin x$$.

Accepted Answer

$${{e}^{x\sin x}}\left[ \cos 2x+\left( \sin x\cos x \right)\left( x\cos x+\sin x \right) \right]$$

Answer

$${{e}^{x\sin x}}\left[ \cos x+\left( \sin x\cos x \right)\left( x\cos x+\sin x \right) \right]$$

Answer

$${{e}^{x\sin x}}\left[ \cos 2x+\left( \sin x+\cos x \right)\left( x\cos x+\sin x \right) \right]$$

Answer

$${{e}^{x\sin x}}\left[ \cos 2x+\left( \sin x\cos x \right)\left( \cos x+x\sin x \right) \right]$$