Logarithmic derivatives Quiz 1

Question 1

Evaluate the value of $$f''\left( x \right)$$ if $$f\left( x \right)=\log \left( {{e}^{x}}+1 \right)$$.

Accepted Answer

$$\dfrac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}}$$

Answer

$$-\dfrac{{{e}^{x}}}{{{\left( {{e}^{x}}+1 \right)}^{2}}}$$

Answer

$$\dfrac{{{e}^{x}}}{{{\left( {{e}^{x}}-1 \right)}^{2}}}$$

Answer

$$-\dfrac{{{e}^{x}}}{{{\left( {{e}^{x}}-1 \right)}^{2}}}$$

Question 2

Evaluate the value of $$f'\left( x \right)$$ if $$f\left( x \right)=\log \left( \dfrac{5x}{{{x}^{2}}+1} \right)$$.

Accepted Answer

$$\dfrac{1-{{x}^{2}}}{x\left( {{x}^{2}}+1 \right)}$$

Answer

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

Answer

$$\dfrac{1-{{x}^{2}}}{{{x}^{2}}\left( {{x}^{2}}+1 \right)}$$

Answer

$$\dfrac{1+{{x}^{2}}}{x\left( {{x}^{2}}-1 \right)}$$

Question 3

Evaluate the value of $$f'\left( x \right)$$ if $$f\left( x \right)=\log \left[ \left( {{x}^{5}}+1 \right)\left( {{x}^{2}}+1 \right) \right]$$.

Accepted Answer

$$\dfrac{7{{x}^{6}}+5{{x}^{2}}+2x}{\left( {{x}^{5}}+1 \right)\left( {{x}^{2}}+1 \right)}$$

Answer

$$\dfrac{3{{x}^{6}}-2{{x}^{2}}+2x}{\left( {{x}^{5}}-1 \right)\left( {{x}^{2}}-1 \right)}$$

Answer

$$\dfrac{3{{x}^{6}}+2{{x}^{2}}+2x}{\left( {{x}^{5}}+1 \right)\left( {{x}^{2}}+1 \right)}$$

Answer

$$\dfrac{4{{x}^{6}}+5{{x}^{2}}+2}{\left( {{x}^{5}}+1 \right)\left( {{x}^{2}}+1 \right)}$$

Question 4

Evaluate the value of $$\dfrac{d\left[ \log \left( {{\sin }^{2}}x{{\cos }^{2}}x \right) \right]}{dx}$$.

Accepted Answer

$$\dfrac{\sin 2x\cos 2x}{{{\sin }^{2}}x{{\cos }^{2}}x}$$

Answer

$$\dfrac{{{\sin }^{2}}x\left( 1-2{{\sin }^{2}}x \right)}{{{\cos }^{2}}x}$$

Answer

$$\dfrac{{{\cos }^{2}}x\left( 1-2{{\sin }^{2}}x \right)}{{{\sin }^{2}}x}$$

Answer

$$\dfrac{\sin 2x\left( 2{{\cos }^{2}}x-1 \right)}{{{\sin }^{2}}x}$$

Question 5

Evaluate the value of $$f'\left( x \right)$$ if $$f\left( x \right)=\log \left( \dfrac{\sin x+\cos x}{\sin x-\cos x} \right)$$.

Accepted Answer

$$\dfrac{-2}{{{\sin }^{2}}x-{{\cos }^{2}}x}$$

Answer

$$\dfrac{\cos 2x}{2{{\cos }^{2}}x-1}$$

Answer

$$\dfrac{\sin 2x}{{{\sin }^{2}}x-{{\cos }^{2}}x}$$

Answer

$$\dfrac{2\sin x\cos x}{{{\cos }^{2}}x-{{\sin }^{2}}x}$$