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