Find the derivative of the following functions.
f:x\longmapsto \log_{3}\left(x\right)
The derivative of a logarithmic function is:
\left(log_{a}\left(x\right)\right)^{'}=\dfrac{1}{x\cdot\ln\left(a\right)}
Here, we have:
f'\left(x\right)=\left(log_{3}\left(x\right)\right)^{'}=\dfrac{1}{x\cdot\ln\left(3\right)}
For any x \in \mathbb{R^*_+} , f is differentiable and we have:
f'\left(x\right)=\dfrac{1}{x\cdot\ln\left(3\right)}
f:x\longmapsto \ln\left(x\right)
The derivative of a logarithmic function is:
\left(log_{a}\left(x\right)\right)^{'}=\dfrac{1}{x\cdot\ln\left(a\right)}
Here, we have:
f'\left(x\right)=\left(\ln\left(x\right)\right)^{'}=\dfrac{1}{x\cdot\ln\left(e\right)}=\dfrac{1}{x}
For any x \in \mathbb{R^*_+} , f is differentiable and we have:
f'\left(x\right)=\dfrac{1}{x}
f:x\longmapsto 5\log\left(x\right)
The derivative of a logarithmic function is:
\left(log_{a}\left(x\right)\right)^{'}=\dfrac{1}{x\cdot\ln\left(a\right)}
Also, if c is a constant and f is a function, then:
\left(c\cdot f\right)'=c\cdot\left(f\right)'
Here, we have:
f'\left(x\right)=\left(5\log\left(x\right)\right)^{'}=5\cdot\left(\log\left(x\right)\right)'=5\cdot\dfrac{1}{x\cdot\ln\left(10\right)}=\dfrac{5}{x\ln\left(10\right)}.
For any x \in \mathbb{R^*_+} , f is differentiable and we have:
f'\left(x\right)=\dfrac{5}{x\ln\left(10\right)}.
f:x\longmapsto 4\log_{2}\left(3x\right)
The derivative of a logarithmic function is:
\left(log_{a}\left(x\right)\right)^{'}=\dfrac{1}{x\cdot\ln\left(a\right)}
Also, if c is a constant and f is a function, then:
\left(c\cdot f\right)'=c\cdot\left(f\right)'
If f is a function then:
\left(log_{a}\left(f\right)\right)^{'}=\dfrac{f'}{f\cdot\ln\left(a\right)}
Here, we have:
f'\left(x\right)=\left(4\log_{2}\left(3x\right)\right)'=4\cdot\left(\log_{2}\left(3x\right)\right)'=4\cdot\dfrac{\left(3x\right)'}{3x\cdot\ln\left(2\right)}=4\cdot\dfrac{3}{3x\cdot\ln\left(2\right)}=\dfrac{4}{x\ln\left(2\right)}.
For any x \in \mathbb{R^*_+} , f is differentiable and we have:
f'\left(x\right)=\dfrac{4}{x\ln\left(2\right)}.
f:x\longmapsto \dfrac{1}{3}\log_{\frac{1}{9}}\left(x\right)
The derivative of a logarithmic function is:
\left(log_{a}\left(x\right)\right)^{'}=\dfrac{1}{x\cdot\ln\left(a\right)}
Also, if c is a constant and f is a function, then:
\left(c\cdot f\right)'=c\cdot\left(f\right)'
Here, we have:
f'\left(x\right)=\left( \dfrac{1}{3}\log_{\frac{1}{9}}\left(x\right)\right)'=\dfrac{1}{3}\cdot\left(\log_{\frac{1}{9}}\left(x\right)\right)'=\dfrac{1}{3}\cdot\dfrac{1}{x\ln\left(\dfrac{1}{9}\right)}=-\dfrac{1}{3x\ln\left(9\right)}
For any x \in \mathbb{R^*_+} , f is differentiable and we have:
f'\left(x\right)=-\dfrac{1}{3x\ln\left(9\right)}.
f:x\longmapsto e^{2}\ln\left(x\right)
The derivative of a logarithmic function is:
\left(log_{a}\left(x\right)\right)^{'}=\dfrac{1}{x\cdot\ln\left(a\right)}
Also, if c is a constant and f is a function, then:
\left(c\cdot f\right)'=c\cdot\left(f\right)'
Here, we have:
f'\left(x\right)=\left(e^{2}\ln\left(x\right)\right)^{'}=e^{2}\cdot\left(\ln\left(x\right)\right)'=e^{2}\cdot\dfrac{1}{x\cdot\ln\left(e\right)}=\dfrac{e^{2}}{x}
For any x \in \mathbb{R^*_+} , f is differentiable and we have:
f'\left(x\right)=\dfrac{e^{2}}{x}.