Find the derivative of the following function. \[y=2\left| -{{\log }_{0.4}}x \right|+7\]

Hint: To find the derivative of the function given in the question, one must start by
 simplifying the given function using properties of logarithmic function and then differentiating the terms given in the function using sum and product rule of differentiation.

To find the derivative of the function\[y=2\left| -{{\log }_{0.4}}x \right|+7\], we will differentiate it with respect to the variable\[x\]using some logarithmic properties.
We will first simplify the given function.
We know that\[{{\log }_{b}}a=\dfrac{\log a}{\log b}\].
Substituting\[a=x,b=0.4\], we have\[y=2\left| -{{\log }_{0.4}}x \right|+7=2\left| \dfrac{-\log x}{\log 0.4} \right|+7\].
We will remove the modulus depending if\[x\]is greater or less than 0.
We know\[\log 0.4<0\].
Case 1: If\[x>1\], we have\[\log x>0\] .Thus, we have\[y=f(x)=\dfrac{-2\log x}{\log 0.4}-7\].
We will use sum rule of differentiation of two functions such that if\[y=f(x)=g(x)+h(x)\]then\[\dfrac{dy}{dx}=\dfrac{dg(x)}{dx}+\dfrac{dh(x)}{dx}\]. \[...(1)\]
We know that differentiation of any function of the form\[y=a\log x+b\]is\[\dfrac{dy}{dx}=\dfrac{a}{x}\].
Substituting\[a=-\dfrac{2}{\log 0.4},b=0\], we have\[\dfrac{dy}{dx}=\dfrac{dg(x)}{dx}=-\dfrac{2}{x\log 0.4}\]. \[...(2)\]
We know that the differentiation of a constant function with respect to any variable is 0.
Thus, we have\[\dfrac{dy}{dx}=\dfrac{dh(x)}{dx}=0\]. \[...(3)\]
Substituting equation\[(2)\]and\[(3)\]in equation\[(1)\], we get\[\dfrac{dy}{dx}=\dfrac{df(x)}{dx}\dfrac{dg(x)}{dx}+\dfrac{dh(x)}{dx}=-\dfrac{2}{x\log 0.4}\].
Thus, differentiation of the function\[y=2\left| -{{\log }_{0.4}}x \right|+7\]is\[\dfrac{dy}{dx}=-\dfrac{2}{x\log 0.4}\].
Case 2: If\[x<1\], we have\[\log x<0\] .Thus, we have\[y=f(x)=\dfrac{2\log x}{\log 0.4}-7\].

We will use sum rule of differentiation of two functions such that if\[y=f(x)=g(x)+h(x)\]then\[\dfrac{dy}{dx}=\dfrac{dg(x)}{dx}+\dfrac{dh(x)}{dx}\]. \[...(4)\]

We know that differentiation of any function of the form\[y=a\log x+b\]is\[\dfrac{dy}{dx}=\dfrac{a}{x}\].
Substituting\[a=\dfrac{2}{\log 0.4},b=0\], we have\[\dfrac{dy}{dx}=\dfrac{dg(x)}{dx}=\dfrac{2}{x\log 0.4}\]. \[...(5)\]
We know that the differentiation of a constant function with respect to any variable is 0.
Thus, we have\[\dfrac{dy}{dx}=\dfrac{dh(x)}{dx}=0\]. \[...(6)\]
Substituting equation\[(5)\]and\[(6)\]in equation\[(4)\], we get\[\dfrac{dy}{dx}=\dfrac{df(x)}{dx}\dfrac{dg(x)}{dx}+\dfrac{dh(x)}{dx}=\dfrac{2}{x\log 0.4}\].
Thus, differentiation of the function\[y=2\left| -{{\log }_{0.4}}x \right|+7\]is\[\dfrac{dy}{dx}=\dfrac{2}{x\log 0.4}\].
Note: The first derivative of any function signifies the slope of the function. Also, we get
 different values of derivatives of the function based on different values of\[x\]. Thus, one
 should remove modulus carefully considering all the cases.