Derivative of Logarithmic Functions

Derivative of Logarithmic Functions

$$ \large \dfrac{d}{dx}\log_e{x} = \dfrac{1}{x} \\
\dfrac{d}{dx}\log_e{f(x)} = \dfrac{1}{f(x)} \times f'(x)$$

Example 1

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{(x^2+1)}$.

\( \begin{align} \displaystyle
\dfrac{dy}{dx} &= \dfrac{d}{dx}\log_e{(x^2+1)} \\
&= \dfrac{1}{x^2+1} \times \dfrac{d}{dx}(x^2+1) \\
&= \dfrac{1}{x^2+1} \times 2x \\
&= \dfrac{2x}{x^2+1} \\
\end{align} \)

Example 2

Find $\displaystyle \dfrac{dy}{dx}$ if $y=x^2\log_e{(2x-1)}$.

\( \begin{align} \displaystyle \require{AMSsymbols} \require{color}
\dfrac{dy}{dx} &= \dfrac{d}{dx}x^2 \times \log_e{(2x-1)} + x^2 \times \dfrac{d}{dx}\log_e{(2x-1)} &\color{red} \text{product rule}\\
&= 2x \times \log_e{(2x-1)} + x^2 \times \dfrac{1}{2x-1} \times \dfrac{d}{dx}{(2x-1)} \\
&= 2x\log_e{(2x-1)} + x^2 \times \dfrac{1}{2x-1} \times 2 \\
&= 2x\log_e{(2x-1)} + \dfrac{2x^2}{2x-1}
\end{align} \)

The laws of logarithmic can help to find the derivative of logarithmic Functions more easily.
$$ \begin{align}
\log_e{(ab)} &= \log_e{a} + \log_e{b} \\
\log_e{\dfrac{a}{b}} &= \log_e{a}-\log_e{b} \\
\log_ea^n &= n\log_e{a} \\
\end{align} $$

Example 3

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{(x^2+1)(x^3-1)}$.

\( \begin{align} \displaystyle
y &= \log_e{(x^2+1)(x^3-1)} \\
&= \log_e{(x^2+1)} + \log_e{(x^3-1)} \\
\dfrac{dy}{dx} &= \dfrac{d}{dx}\log_e{(x^2+1)} + \dfrac{d}{dx}\log_e{(x^3-1)} \\
&= \dfrac{1}{x^2+1}\times \dfrac{d}{dx}{(x^2+1)} + \dfrac{1}{x^3-1} \times \dfrac{d}{dx}(x^3-1) \\
&= \dfrac{1}{x^2+1}\times 2x + \dfrac{1}{x^3-1} \times 3x^2 \\
&= \dfrac{2x}{x^2+1} + \dfrac{3x^2}{x^3-1}
\end{align} \)

Example 4

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{\sqrt{x^2+2}}$.

\( \begin{align} \displaystyle \require{color}
y &= \log_e{\sqrt{x^2+2}} \\
&= \log_e{(x^2+2)^{\frac{1}{2}}} \\
&= \dfrac{1}{2}\log_e{(x^2+2)} \\
\dfrac{dy}{dx} &= \dfrac{d}{dx}\dfrac{1}{2}\log_e{(x^2+2)} \\
&= \dfrac{1}{2} \times \dfrac{1}{x^2+2} \times \dfrac{d}{dx}(x^2+2) \\
&= \dfrac{1}{2} \times \dfrac{1}{x^2+2} \times 2x \\
&= \dfrac{x}{x^2+2}
\end{align} \)

Example 5

Find $\displaystyle \dfrac{dy}{dx}$ if $y=\log_e{\dfrac{x^2}{(x+3)(x-5)}}$.

\( \begin{align} \displaystyle \require{color}
y &= \log_e{\dfrac{x^2}{(x+3)(x-5)}} \\
&= \log_e{x^2}-\log_e{(x+3)}-\log_e{(x-5)} \\
&= 2\log_e{x}-\log_e{(x+3)}-\log_e{(x-5)} \\
\dfrac{dy}{dx} &= \dfrac{d}{dx}2\log_e{x}-\dfrac{d}{dx}\log_e{(x+3)}-\dfrac{d}{dx}\log_e{(x-5)} \\
&= \dfrac{2}{x}-\dfrac{1}{x+3}-\dfrac{1}{x-5}
\end{align} \)

Example 6

Compare the derivatives of $\log_e{x^3}$ and $(\log_e{x})^3$.

\( \begin{align} \displaystyle \require{color}
\dfrac{d}{dx}\log_e{x^3} &= \dfrac{d}{dx}3\log_e{x} \\
&= \dfrac{3}{x} \\
\dfrac{d}{dx}(\log_e{x})^3 &= 3(\log_e{x})^{3-1} \times \dfrac{d}{dx}\log_e{x} \\
&= 3(\log_e{x})^{2} \times \dfrac{1}{x} \\
&= \dfrac{3(\log_e{x})^{2}}{x}
\end{align} \)

Extension Examples

These Extension Examples require to have some prerequisite skills, including;
\( \begin{align} \displaystyle
\dfrac{d}{dx}\sin{x} &= \cos{x} \\
\dfrac{d}{dx}\cos{x} &= -\sin{x} \\
\dfrac{d}{dx}e^x &= e^x \\
\end{align} \)

Example 7

Find $\displaystyle \dfrac{dy}{dx}$ of $e^x\log_e{x}$, known that $\dfrac{d}{dx}e^x = e^x$.

\( \begin{align} \displaystyle
\dfrac{dy}{dx} &= \dfrac{d}{dx}e^x \times \log_e{x} + e^x \times \dfrac{d}{dx}\log_e{x} \\
&= e^x \times \log_e{x} + e^x \times \dfrac{1}{x} \\
&= e^x\log_e{x} + \dfrac{e^x}{x}
\end{align} \)

Example 8

Find $\displaystyle \dfrac{dy}{dx}$ of $\displaystyle y=\log_e{\dfrac{\sin{x}}{\cos{x}}}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$ and $\dfrac{d}{dx}\cos{x} = -\sin{x}$.

\( \begin{align} \displaystyle
y &= \log_e{\dfrac{\sin{x}}{\cos{x}}} \\
&= \log_e{\sin{x}}-\log_e{\cos{x}} \\
\dfrac{dy}{dx} &= \dfrac{d}{dx}\log_e{\sin{x}}-\dfrac{d}{dx}\log_e{\cos{x}} \\
&= \dfrac{1}{\sin{x}} \times \dfrac{d}{dx}\sin{x}-\dfrac{1}{\cos{x}} \times \dfrac{d}{dx}\cos{x}\\
&= \dfrac{1}{\sin{x}} \times \cos{x}-\dfrac{1}{\cos{x}} \times (-\sin{x})\\
&= \dfrac{\cos{x}}{\sin{x}} + \dfrac{\sin{x}}{\cos{x}}
\end{align} \)

Example 9

Find $\displaystyle \dfrac{dy}{dx}$ of $\displaystyle y=\log_e{\sin{x^2}}$, known that $\dfrac{d}{dx}\sin{x} = \cos{x}$.

\( \begin{align} \displaystyle
\dfrac{d}{dx} &= \dfrac{d}{dx}\log_e{\sin{x^2}} \\
&= \dfrac{1}{\sin{x^2}} \times \dfrac{d}{dx} \sin{x^2} \\
&= \dfrac{1}{\sin{x^2}} \times \cos{x^2} \times \dfrac{d}{dx}x^2\\
&= \dfrac{1}{\sin{x^2}} \times \cos{x^2} \times 2x \\
&= \dfrac{2x\cos{x^2}}{\sin{x^2}}
\end{align} \)

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