Find the derivative of a function of the form x -> f(x)/g(x) - High school Calculus Exercises

Find the derivative of the following function.

f:x\longmapsto \dfrac{\cos\left(x\right)}{2^x}

f'\left(x\right)=\dfrac{-\sin{\left(x\right)}}{\ln{4^x}}

f'\left(x\right)=\dfrac{-\sin{\left(x\right)}}{2^x\ln{\left(2\right)}}

f'\left(x\right)= -\dfrac{2^x\left(\sin{\left(x\right)} + \cos{\left(x\right)} \cdot \ln{\left(2\right)}\right)}{4^x}

f'\left(x\right)= -\dfrac{\sin{\left(x\right)} + \cos{\left(x\right)} \cdot \ln{\left(2\right)}}{2^x}

f:x \longmapsto \dfrac{ \sin{\left(x\right)} } { \cos{\left(x\right)} }

f'\left(x\right)=-\dfrac{\cos{\left(x\right)}} {\sin{\left(x\right)}}

f'\left(x\right)=\dfrac{\cos^2{\left(x\right)}-\sin^2{\left(x\right)}}{\cos^2{\left(x\right)}}

f'\left(x\right)= \dfrac{1}{\cos^2{\left(x\right)}}

f'\left(x\right)= 1 + \sin^2{\left(x\right)}

f:x \longmapsto \dfrac{ \log_3{\left(x\right)} } {\sin{\left(x\right)}}

f'\left(x\right) = \dfrac{\sin\left(x\right)- x\ln\left(x\right)\cos{\left(x\right)}}{x\cdot\sin^2{\left(x\right)} \cdot \ln{3}}

f'\left(x\right) = \dfrac{x\ln{3}}{\sin{\left(x\right)}} + \dfrac{\log_3{\left(x\right)}\cdot \cos{\left(x\right)}}{\sin^2{\left(x\right)}}

f'\left(x\right) = \dfrac{1}{x\cdot\sin{\left(x\right)} \cdot \ln{3}} - \log_3{\left(x\right)}\cdot \cos{\left(x\right)}

f'\left(x\right)= \dfrac{x\ln{3}}{\cos{\left(x\right)}}

f:x \longmapsto \dfrac{e^{x} } {3^x}

f'\left(x\right) = \dfrac{ e^x \left(1 - \ln{3}\right)}{2}

f'\left(x\right) = \dfrac{ e^x \left(1 - \ln{3}\right)}{3^{x}}

f'\left(x\right) = \dfrac{ e^x }{3^{x}}

f'\left(x\right) = -\dfrac{ e^x \left(1 + \ln{3}\right)}{3^{x}}

f:x \longmapsto \dfrac{\ln{x} } {\log_2{x}}

f'\left(x\right) = -\dfrac{1}{\ln{2}}

f'\left(x\right) = \ln{2}

f'\left(x\right) = 1-\dfrac{1}{ \ln{2}\cdot \ln{\left(x\right)}\cdot \log_2{\left(x\right)} }

f'\left(x\right) = 0

f:x \longmapsto \dfrac{\ln{x} } {4^x}

f'\left(x\right) =\dfrac{1}{4^x} \left(\dfrac{1}{x} - \ln{x} \cdot \ln{4}\right)

f'\left(x\right) =\dfrac{1}{4} - \ln{x} \cdot \ln{4}

f'\left(x\right) =\dfrac{1}{4^x} \left(\dfrac{1}{x} - \ln{4x}\right)

f'\left(x\right) = \dfrac{1}{ x}\cdot\dfrac{1}{4^x\ln{4}}

f:x \longmapsto \dfrac{1/x } {\ln{\left(x\right)}}

f'\left(x\right)= -\dfrac{1}{x^2}\left(\dfrac{1}{\ln{\left(x\right)}}\right)

f'\left(x\right)= -\dfrac{1}{\ln{\left(x^2\right)}}

f'\left(x\right)= -\dfrac{1}{x^2}\left(\dfrac{ \ln{\left(x\right)} +1}{\ln^2{\left(x\right)}}\right)

f'\left(x\right)= \dfrac{1}{x^2}\left(\dfrac{1}{ \ln{\left(x\right)}}\right)