Generating...                               s20quiz1315_n16

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

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

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

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

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

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

  4. Find the derivative $f'(x)$ for $f(x) =\displaystyle \arccos e^ {- 2\,x } $ .

    $f'(x) =\displaystyle -{{1}\over{\sqrt{1-e^ {- 4\,x }}}} $ $f'(x) =\displaystyle {{2\,e^ {- 2\,x }}\over{\sqrt{1-e^ {- 4\,x }}}} $ $f'(x) =\displaystyle -{{2\,e^ {- 2\,x }}\over{\cos e^ {- 2\,x }}} $ $f'(x) =\displaystyle {{e^ {- 2\,x }}\over{\cos e^ {- 2\,x }}} $ $f'(x) =\displaystyle -{{2\,e^ {- 2\,x }\,\sin e^ {- 2\,x }}\over{\cos ^2e^ {- 2\,x }}} $ $f'(x) =\displaystyle -{{e^ {- 2\,x }}\over{\sqrt{1-e^ {- 4\,x }}}} $

  5. Find the derivative $f'(x)$ for $f(x) =\displaystyle \ln \cosh 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. If $\displaystyle \cos y+x^2\,y^2=y $ , then find $\displaystyle \frac{dy}{dx}$ by implicit differentiation.

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

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

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

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

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

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

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

  10. Find the derivative $f'(x)$ for $f(x) =\displaystyle \ln \left\vert x^2-x-3\right\vert $ .

    $f'(x) =\displaystyle 2\,x-1 $ $f'(x) =\displaystyle \left(2\,x-1\right)\,\ln \left\vert x^2-x-3\right\vert $ $f'(x) =\displaystyle {{2\,x-1}\over{x^2-x-3}} $ $f'(x) =\displaystyle {{1}\over{x^2-x-3}} $


Department of Mathematics
Last modified: 2026-02-06