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Derivative of $$$\ln\left(\frac{a^{2}}{x^{2}}\right)$$$ with respect to $$$x$$$
Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps
Your Input
Find $$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right)$$$.
Solution
The function $$$\ln\left(\frac{a^{2}}{x^{2}}\right)$$$ is the composition $$$f{\left(g{\left(x \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \ln\left(u\right)$$$ and $$$g{\left(x \right)} = \frac{a^{2}}{x^{2}}$$$.
Apply the chain rule $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)\right)}$$The derivative of the natural logarithm is $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)$$Return to the old variable:
$$\frac{\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)}{{\color{red}\left(\frac{a^{2}}{x^{2}}\right)}}$$Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = a^{2}$$$ and $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:
$$\frac{x^{2} {\color{red}\left(\frac{d}{dx} \left(\frac{a^{2}}{x^{2}}\right)\right)}}{a^{2}} = \frac{x^{2} {\color{red}\left(a^{2} \frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)}}{a^{2}}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = -2$$$:
$$x^{2} {\color{red}\left(\frac{d}{dx} \left(\frac{1}{x^{2}}\right)\right)} = x^{2} {\color{red}\left(- \frac{2}{x^{3}}\right)}$$Thus, $$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right) = - \frac{2}{x}$$$.
Answer
$$$\frac{d}{dx} \left(\ln\left(\frac{a^{2}}{x^{2}}\right)\right) = - \frac{2}{x}$$$A