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This is one of our many quizzes on Product rule. Click on this to access more such competitive quizzes on Differentiation.

Product rule Quiz 2

mathematics

 First use the product rule, $\dfrac{d}{dx}(uv)=u\dfrac{d}{dx}v+v\dfrac{d}{dx}u$ and $\dfrac{d}{dx}(\log x)=\dfrac{1}{x}$.

  $ y={{e}^{abx}}\log x \\  \Rightarrow y'={{e}^{abx}}\dfrac{d}{dx}\left( \log x \right)+\log x\dfrac{d}{dx}\left( {{e}^{abx}} \right) \\  \Rightarrow y'={{e}^{abx}}\dfrac{1}{x}+\log x\times ab{{e}^{abx}} \\  \Rightarrow y'=\dfrac{{{e}^{abx}}}{x}+ab{{e}^{abx}}\log x=\dfrac{{{e}^{abx}}}{x}+aby \\ $

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

 First use the product rule, $\dfrac{d}{dx}(uv)=u\dfrac{d}{dx}v+v\dfrac{d}{dx}u$, then the derivatives of the trigonometric ratios and $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$.

  $  f(x)=\left( {{\sin }^{4}}x+4{{x}^{2}} \right)\left( 2{{\cos }^{4}}x+7x \right) \\  \Rightarrow f'(x)=\left( {{\sin }^{4}}x+4{{x}^{2}} \right)\dfrac{d}{dx}\left( 2{{\cos }^{4}}x+7x \right)+\left( 2{{\cos }^{4}}x+7x \right)\dfrac{d}{dx}\left( {{\sin }^{4}}x+4{{x}^{2}} \right) \\  \Rightarrow f'(x)=\left( {{\sin }^{4}}x+4{{x}^{2}} \right)\left\{ 8{{\cos }^{3}}x\dfrac{d}{dx}\left( \cos x \right)+7 \right\}+\left( 2{{\cos }^{4}}x+7x \right)\left\{ 4{{\sin }^{3}}x\dfrac{d}{dx}\left( \sin x \right)+8x \right\} \\  \Rightarrow f'(x)=\left( {{\sin }^{4}}x+4{{x}^{2}} \right)\left( 7-8\sin x{{\cos }^{3}}x \right)+\left( 2{{\cos }^{4}}x+7x \right)\left( 4\cos x{{\sin }^{3}}x+8x \right) \\ $

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

 First use the product rule, $\dfrac{d}{dx}(uv)=u\dfrac{d}{dx}v+v\dfrac{d}{dx}u$, then the derivatives of the trigonometric ratios and$\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$.

$  y=\dfrac{4}{{{x}^{3}}}\left( 7\sin x+2{{x}^{-2}} \right) \\  \Rightarrow y'=\dfrac{4}{{{x}^{3}}}\dfrac{d}{dx}\left( 7\sin x+2{{x}^{-2}} \right)+\left( 7\sin x+2{{x}^{-2}} \right)\dfrac{d}{dx}\left( \dfrac{4}{{{x}^{3}}} \right) \\  \Rightarrow y'=\dfrac{4}{{{x}^{3}}}\left( 7\cos x-\dfrac{4}{{{x}^{3}}} \right)+\left( 7\sin x+2{{x}^{-2}} \right)\left( \dfrac{-12}{{{x}^{4}}} \right) \\  \Rightarrow y'=\dfrac{4}{{{x}^{4}}}\left[ x\left( 7\cos x-\dfrac{4}{{{x}^{3}}} \right)-3\left( 7\sin x+2{{x}^{-2}} \right) \right] \\ $

 Find $y'$, if $y=\dfrac{1}{\sqrt{x}}{{\left( \log x \right)}^{3}}.$

 $  y=\dfrac{1}{\sqrt{x}}{{\left( \log x \right)}^{3}} \\  \Rightarrow y'=\dfrac{1}{\sqrt{x}}\dfrac{d}{dx}{{\left( \log x \right)}^{3}}+{{\left( \log x \right)}^{3}}\dfrac{d}{dx}\left( \dfrac{1}{\sqrt{x}} \right) \\  \Rightarrow y'=\dfrac{3}{\sqrt{x}}{{\left( \log x \right)}^{2}}\dfrac{d}{dx}\left( \log x \right)+{{\left( \log x \right)}^{3}}\left( -\dfrac{1}{2} \right)\left( \dfrac{1}{x\sqrt{x}} \right) \\  \Rightarrow y'=\dfrac{3}{x\sqrt{x}}{{\left( \log x \right)}^{2}}-{{\left( \log x \right)}^{3}}\left( \dfrac{1}{2x\sqrt{x}} \right) \\  \Rightarrow y'=\dfrac{{{\left( \log x \right)}^{2}}}{x\sqrt{x}}\left( 3-\dfrac{1}{2}\log x \right) \\ $

 Find $y'$, if $y=4{{e}^{x{{e}^{x}}}}{{\sin }^{2}}x\tan x.$

 First use the product rule, $\dfrac{d}{dx}(uv)=u\dfrac{d}{dx}v+v\dfrac{d}{dx}u$, then the derivatives of the trigonometric ratios, $\dfrac{d}{dx}{{e}^{x}}={{e}^{x}}$ and $\dfrac{d}{dx}{{x}^{n}}=n{{x}^{n-1}}$.

 $  y=4{{e}^{x{{e}^{x}}}}{{\sin }^{2}}x\tan x \\  \Rightarrow y'=4{{e}^{x{{e}^{x}}}}\dfrac{d}{dx}\left( {{\sin }^{2}}x\tan x \right)+\left( {{\sin }^{2}}x\tan x \right)\dfrac{d}{dx}\left( 4{{e}^{x{{e}^{x}}}} \right) \\  \Rightarrow y'=4{{e}^{x{{e}^{x}}}}\left[ {{\sin }^{2}}x\dfrac{d}{dx}\left( \tan x \right)+\tan x\dfrac{d}{dx}\left( {{\sin }^{2}}x \right) \right]+\left( {{\sin }^{2}}x\tan x \right)4{{e}^{x{{e}^{x}}}}\dfrac{d}{dx}\left( x{{e}^{x}} \right) \\  \Rightarrow y'=4{{e}^{x{{e}^{x}}}}\left[ {{\sin }^{2}}x{{\sec }^{2}}x+2\tan x\sin x\dfrac{d}{dx}\left( \sin x \right) \right]+\left( {{\sin }^{2}}x\tan x \right)4{{e}^{x{{e}^{x}}}}\left[ x\dfrac{d}{dx}\left( {{e}^{x}} \right)+{{e}^{x}}\dfrac{d}{dx}\left( x \right) \right] \\  \Rightarrow y'=4{{e}^{x{{e}^{x}}}}\left[ {{\sin }^{2}}x{{\sec }^{2}}x+2\left( \dfrac{\sin x}{\cos x} \right)\sin x\cos x \right]+\left( {{\sin }^{2}}x\tan x \right)4{{e}^{x{{e}^{x}}}}\left[ x\dfrac{d}{dx}\left( {{e}^{x}} \right)+{{e}^{x}}\dfrac{d}{dx}\left( x \right) \right] \\  \Rightarrow y'=4{{e}^{x{{e}^{x}}}}\left[ \left( {{\sin }^{2}}x{{\sec }^{2}}x+2{{\sin }^{2}}x \right)+\left( {{\sin }^{2}}x\tan x \right)\left( x{{e}^{x}}+{{e}^{x}} \right) \right] \\  \Rightarrow y'=4{{e}^{x{{e}^{x}}}}{{\sin }^{2}}x\left[ {{\sec }^{2}}x+2+{{e}^{x}}(x+1)\tan x \right] \\ $