Generating...                               s20quiz1315_n14

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

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

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

    $\displaystyle \frac{dy}{dx} = -{{\cosh y+3\,x^3\,y^2}\over{3\,x^2\,y^3-1}} $ $\displaystyle \frac{dy}{dx} = \sinh y+x^3\,y^3 $ $\displaystyle \frac{dy}{dx} = -{{3\,x^2\,y^3-1}\over{\cosh y+3\,x^3\,y^2}} $ $\displaystyle \frac{dy}{dx} = -{{3\,x^2\,y^3}\over{\cosh y+3\,x^3\,y^2-1}} $ $\displaystyle \frac{dy}{dx} = -{{\cosh y+3\,x^3\,y^2-1}\over{3\,x^2\,y^3}} $ $\displaystyle \frac{dy}{dx} = 3\,x^2\,y^3 $

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

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

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

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

    $\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{y}}\over{2\,\sqrt{x}\,\sqrt{y}-\sqrt{x}}} $ $\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{x}}\over{\left(2\,\sqrt{x}-1\right)\,\sqrt{y}}} $

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

    $\displaystyle y' = y \left( {{3\,x^2}\over{x^3+4}}+{{2\,x}\over{x^2+4}}+{{1}\over{x-4}} \right) $ $\displaystyle y' = y \left( -{{6\,x^2}\over{x^3+4}}-{{2\,x}\over{x^2+4}}+{{2}\over{x-4}} \right) $ $\displaystyle y' = y \left( -{{6\,x^2}\over{x^3+4}}+{{2\,x}\over{x^2+4}}+{{2}\over{x-4}} \right) $ $\displaystyle y' = y \left( -{{6\,x^2}\over{x^3+4}}-{{2\,x}\over{x^2+4}}+{{1}\over{x-4}} \right) $ $\displaystyle y' = y \left( -{{6\,x^2}\over{\left(x^3+4\right)^3}}-{{2\,x}\over{\left(x^2+4 \right)^2}}+2\,\left(x-4\right) \right) $ $\displaystyle y' = y \left( {{3\,x^2}\over{x^3+4}}-{{2\,x}\over{x^2+4}}+{{2}\over{x-4}} \right) $

  8. Find the derivative $f'(x)$ for $f(x) =\displaystyle \arccos x^2 $ .

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

  9. Find the derivative $f'(x)$ for $f(x) =\displaystyle \ln \left\vert x^2+2\,x-1\right\vert $ .

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

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

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


Department of Mathematics
Last modified: 2025-10-30