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**Hint:**To solve this problem, we will need to know the differentiation of composite functions of the form \[y=f\left( g\left( x \right) \right)\]. The derivative of the composite functions \[\dfrac{dy}{dx}\] is evaluated as \[y=\dfrac{df\left( g\left( x \right) \right)}{d\left( g\left( x \right) \right)}\times \dfrac{dg\left( x \right)}{dx}\]. For this problem, we have the composite function of three different functions. We will use a similar method to differentiate it.

**Complete step-by-step solution:**

We are given that \[f(x)=\sin (\log x)\] and \[y=f\left( \dfrac{2x+3}{3-2x} \right)\], we are asked to evaluate \[\dfrac{dy}{dx}\]. We get the expression for y as, \[y=\sin \left( \log \left( \dfrac{2x+3}{3-2x} \right) \right)\]. As we can see that y is a composite function of the form \[f\left( g\left( h\left( x \right) \right) \right)\]. Here, \[f\left( x \right)=\sin x\], \[g\left( x \right)=\log x \And h\left( x \right)=\dfrac{2x+3}{3-2x}\].

As we know that the derivative of \[\sin x\] with respect to x is \[\cos x\]. Thus, the derivatives of \[f\left( g\left( h\left( x \right) \right) \right)\] with respect to \[g\left( h\left( x \right) \right)\] is \[\dfrac{df\left( g\left( h\left( x \right) \right) \right)}{dg\left( h\left( x \right) \right)}=\cos \left( \log \left( \dfrac{2x+3}{3-2x} \right) \right)\].

The derivative of \[\log x\] with respect to x is \[\dfrac{1}{x}\]. Thus, the derivative of \[g\left( h\left( x \right) \right)\] with respect to \[h\left( x \right)\] is \[\dfrac{dg\left( h\left( x \right) \right)}{dh\left( x \right)}=\dfrac{1}{\dfrac{2x+3}{3-2x}}=\dfrac{3-2x}{2x+3}\]

The derivative of \[\dfrac{3-2x}{2x+3}\] with respect to x is \[\dfrac{dh(x)}{dx}=\dfrac{2(3-2x)-(-2)(2x+3)}{{{\left( 3-2x \right)}^{2}}}\], simplifying this expression, we get \[\dfrac{dh(x)}{dx}=\dfrac{12}{{{\left( 3-2x \right)}^{2}}}\]

Using the derivative of a composite function, we can differentiate the function \[y=\sin \left( \log \left( \dfrac{2x+3}{3-2x} \right) \right)\] as,

\[\dfrac{dy}{dx}=\dfrac{df\left( g\left( h\left( x \right) \right) \right)}{dg\left( h\left( x \right) \right)}\times \dfrac{dg\left( h\left( x \right) \right)}{dh\left( x \right)}\times \dfrac{dh(x)}{dx}\]

Substituting the values of the derivatives in the above expression, we get

\[\Rightarrow \dfrac{dy}{dx}=\cos \left( \log \left( \dfrac{2x+3}{3-2x} \right) \right)\times \dfrac{3-2x}{2x+3}\times \dfrac{12}{{{\left( 3-2x \right)}^{2}}}\]

Simplifying the above expression, we get

\[\Rightarrow \dfrac{dy}{dx}=\cos \left( \log \left( \dfrac{2x+3}{3-2x} \right) \right)\dfrac{12}{\left( 3-2x \right)\left( 2x+3 \right)}\]

\[\Rightarrow \dfrac{dy}{dx}=\cos \left( \log \left( \dfrac{2x+3}{3-2x} \right) \right)\dfrac{12}{9-4{{x}^{2}}}\]

Thus, the derivative of the given expression is \[\dfrac{dy}{dx}=\cos \left( \log \left( \dfrac{2x+3}{3-2x} \right) \right)\dfrac{12}{9-4{{x}^{2}}}\].

**Hence, the answer is an option \[\left( B \right)\].**

**Note:**We can use this method to find the derivative of a composite function having more functions than this. In general words, \[y=f\left( g(h(......(x))) \right)\] can be differentiated by differentiating the function outside with respect to the function inside it.

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