Generating...                               s20quiz1315_n20

  1. Find the derivative $f'(x)$ for $f(x) =\displaystyle \arctan e^ {- x } $ .

    $f'(x) =\displaystyle -{{e^ {- x }}\over{e^ {- 2\,x }+1}} $ $f'(x) =\displaystyle {{e^ {- x }}\over{e^ {- 2\,x }+1}} $ $f'(x) =\displaystyle -{{e^ {- x }}\over{\tan e^ {- x }}} $ $f'(x) =\displaystyle {{e^ {- x }}\over{\tan e^ {- x }}} $ $f'(x) =\displaystyle {{1}\over{e^ {- 2\,x }+1}} $ $f'(x) =\displaystyle {{e^ {- x }\,\left(\sec e^ {- x }\right)^2}\over{\tan ^2e^ {- x }}} $

  2. If $\displaystyle y^4+x^4=1 $ , then find $\displaystyle \frac{dy}{dx}$ by implicit differentiation.

    $\displaystyle \frac{dy}{dx} = -{{4\,x^3-1}\over{4\,y^3}} $ $\displaystyle \frac{dy}{dx} = -{{4\,y^3}\over{4\,x^3-1}} $ $\displaystyle \frac{dy}{dx} = y^4+x^4 $ $\displaystyle \frac{dy}{dx} = -{{x^3}\over{y^3}} $ $\displaystyle \frac{dy}{dx} = -{{y^3}\over{x^3}} $ $\displaystyle \frac{dy}{dx} = 4\,x^3 $

  3. If $\displaystyle \cosh y+x^2\,y^2=y $ , then find $\displaystyle \frac{dy}{dx}$ by implicit differentiation.

    $\displaystyle \frac{dy}{dx} = -{{\sinh y+2\,x^2\,y}\over{2\,x\,y^2-1}} $ $\displaystyle \frac{dy}{dx} = -{{2\,x\,y^2-1}\over{\sinh y+2\,x^2\,y}} $ $\displaystyle \frac{dy}{dx} = -{{\sinh y+2\,x^2\,y-1}\over{2\,x\,y^2}} $ $\displaystyle \frac{dy}{dx} = \cosh y+x^2\,y^2 $ $\displaystyle \frac{dy}{dx} = -{{2\,x\,y^2}\over{\sinh y+2\,x^2\,y-1}} $ $\displaystyle \frac{dy}{dx} = 2\,x\,y^2 $

  4. If $\displaystyle \sqrt{y}+\sqrt{x}=x $ , then find $\displaystyle \frac{dy}{dx}$ by implicit differentiation.

    $\displaystyle \frac{dy}{dx} = \sqrt{y}+\sqrt{x} $ $\displaystyle \frac{dy}{dx} = {{2\,\sqrt{x}\,\sqrt{y}-\sqrt{x}}\over{\sqrt{y}}} $ $\displaystyle \frac{dy}{dx} = {{\sqrt{y}}\over{2\,\sqrt{x}\,\sqrt{y}-\sqrt{x}}} $ $\displaystyle \frac{dy}{dx} = {{\left(2\,\sqrt{x}-1\right)\,\sqrt{y}}\over{\sqrt{x}}} $ $\displaystyle \frac{dy}{dx} = {{1}\over{2\,\sqrt{x}}} $ $\displaystyle \frac{dy}{dx} = {{\sqrt{x}}\over{\left(2\,\sqrt{x}-1\right)\,\sqrt{y}}} $

  5. Find the derivative $f'(x)$ for $f(x) =\displaystyle \ln \sinh x $ .

    $f'(x) =\displaystyle {{1}\over{e^{x}-e^ {- x }}} $ $f'(x) =\displaystyle {{1}\over{e^{x}+e^ {- x }}} $ $f'(x) =\displaystyle {{e^{x}-e^ {- x }}\over{e^{x}+e^ {- x }}} $ $f'(x) =\displaystyle {{e^{x}+e^ {- x }}\over{e^{x}-e^ {- x }}} $

  6. Let $\displaystyle y={{\left(x^2+4\right)^4\,\left(x^4+1\right)^3}\over{\sqrt{x-4}}} $ . Find the derivative in the form $y' = y (\cdots)$ by logarithmic differentiation.

    $\displaystyle y' = y \left( {{4\,x^3}\over{x^4+1}}+{{8\,x}\over{x^2+4}}-{{1}\over{2\,\left(x-4 \right)}} \right) $ $\displaystyle y' = y \left( 12\,x^3\,\left(x^4+1\right)^2+8\,x\,\left(x^2+4\right)^3-{{1}\over{ 2\,\left(x-4\right)^{{{3}\over{2}}}}} \right) $ $\displaystyle y' = y \left( {{12\,x^3}\over{x^4+1}}+{{2\,x}\over{x^2+4}}-{{1}\over{2\,\left(x-4 \right)}} \right) $ $\displaystyle y' = y \left( {{12\,x^3}\over{x^4+1}}+{{8\,x}\over{x^2+4}}+{{1}\over{x-4}} \right) $ $\displaystyle y' = y \left( {{12\,x^3}\over{x^4+1}}+{{8\,x}\over{x^2+4}}-{{1}\over{2\,\left(x-4 \right)}} \right) $ $\displaystyle y' = y \left( {{4\,x^3}\over{x^4+1}}+{{2\,x}\over{x^2+4}}+{{1}\over{x-4}} \right) $

  7. Find the derivative $f'(x)$ for $f(x) =\displaystyle x^{x} $ .

    $f'(x) =\displaystyle x^{x}\,\ln x+x^{x} $ $f'(x) =\displaystyle x^{x} $ $f'(x) =\displaystyle \left(\ln x\right)^{x}\,\left(\ln \ln x+{{1}\over{\ln x}} \right) $ $f'(x) =\displaystyle \left(\ln x\right)^{x-1} $ $f'(x) =\displaystyle \left(\ln x\right)^{x}\,\ln \ln x $

  8. Find the derivative $f'(x)$ for $f(x) =\displaystyle \left(\ln x\right)^{x} $ .

    $f'(x) =\displaystyle x^{x} $ $f'(x) =\displaystyle \left(\ln x\right)^{x}\,\ln \ln x+\left(\ln x\right)^{x-1} $ $f'(x) =\displaystyle \left(\ln x\right)^{x-1} $ $f'(x) =\displaystyle \left(\ln x\right)^{x}\,\ln \ln x $ $f'(x) =\displaystyle x^{x}\,\left(\ln x+1\right) $

  9. Find the derivative $f'(x)$ for $f(x) =\displaystyle \ln \left\vert 2\,x^2-4\,x-5\right\vert $ .

    $f'(x) =\displaystyle 4-4\,x $ $f'(x) =\displaystyle {{1}\over{-2\,x^2+4\,x+5}} $ $f'(x) =\displaystyle \left(4-4\,x\right)\,\ln \left\vert 2\,x^2-4\,x-5\right\vert $ $f'(x) =\displaystyle {{4\,x-4}\over{2\,x^2-4\,x-5}} $

  10. If $\displaystyle e^{x^2\,y}=y+x $ , then find $\displaystyle \frac{dy}{dx}$ by implicit differentiation.

    $\displaystyle \frac{dy}{dx} = -{{x^2\,e^{x^2\,y}-1}\over{2\,x\,y\,e^{x^2\,y}-1}} $ $\displaystyle \frac{dy}{dx} = -{{2\,x\,y\,e^{x^2\,y}}\over{x^2\,e^{x^2\,y}-1}} $ $\displaystyle \frac{dy}{dx} = -{{2\,x\,y\,e^{x^2\,y}-1}\over{x^2\,e^{x^2\,y}-1}} $ $\displaystyle \frac{dy}{dx} = e^{x^2\,y} $ $\displaystyle \frac{dy}{dx} = -{{e^ {- x^2\,y }\,\left(x^2\,e^{x^2\,y}-1\right)}\over{2\,x\,y}} $ $\displaystyle \frac{dy}{dx} = 2\,x\,y\,e^{x^2\,y} $


Department of Mathematics
Last modified: 2026-07-16