Finding the Nth Derivatives of Various Functions
<p>\textbf{(a) } \frac{d^n}{dx^n} \left( \frac{x^{n+1}}{x^2+7x+12} \right)</p>
<p>\text{As this is a rational function, the nth derivative would typically require applying the quotient rule repeatedly, which is complex and not straightforward without a specific value for } n.</p>
<p>\textbf{(b) } \frac{d^n}{dx^n} ( \cos x \cdot \cos 2x \cdot \cos 3x)</p>
<p>\text{This product of trigonometric functions would require using the product rule iteratively in combination with trigonometric identities.}</p>
<p>\textbf{(c) } \frac{d^n}{dx^n} (\sin x^{2} \cdot \cos x)</p>
<p>\text{Applying Leibniz's rule for differentiation of the product of two functions multiple times would be needed here.}</p>
<p>\textbf{(d)}\text{ This part of the question is incomplete as the function to differentiate is not provided.}</p>
<p>\text{For parts (b) and (c), without additional information on the value of } n, \text{ it is not feasible to provide a general nth derivative. Additionally, part (d) lacks the necessary information to perform any derivative.}</p>
