Derivative of $$$x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}$$$ with respect to $$$x$$$
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Find $$$\frac{d}{dx} \left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right)$$$.
Solution
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 $$$c = \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}$$$ and $$$f{\left(x \right)} = x^{2}$$$:
$${\color{red}\left(\frac{d}{dx} \left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right)\right)} = {\color{red}\left(\cos^{2}{\left(\tanh{\left(\eta \right)} \right)} \frac{d}{dx} \left(x^{2}\right)\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:
$$\cos^{2}{\left(\tanh{\left(\eta \right)} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = \cos^{2}{\left(\tanh{\left(\eta \right)} \right)} {\color{red}\left(2 x\right)}$$Thus, $$$\frac{d}{dx} \left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right) = 2 x \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}$$$.
Answer
$$$\frac{d}{dx} \left(x^{2} \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}\right) = 2 x \cos^{2}{\left(\tanh{\left(\eta \right)} \right)}$$$A