Differentiate the following: y=(7x^2+2)sinx

1. To differentiate the function \( y = (7x^2 + 2) \sin x \), one must apply the product rule of differentiation, which is essential in A Level Maths. The product rule states that if you have a function that is the product of two differentiable functions, say \( u \) and \( v \), then the derivative of their product is given by \( u’v + uv’ \). In this case, we can identify \( u = 7x^2 + 2 \) and \( v = \sin x \). 

2. First, we need to find the derivatives of both components. The derivative of \( u \), which is \( 7x^2 + 2 \), is calculated as \( u’ = 14x \). Meanwhile, the derivative of \( v = \sin x \) is \( v’ = \cos x \). With these derivatives in hand, we can now apply the product rule to find the derivative of the entire function \( y \). Substituting the values into the product rule formula, we have \( y’ = u’v + uv’ = (14x)(\sin x) + (7x^2 + 2)(\cos x) \).

3. Therefore, the final expression for the derivative of the function \( y = (7x^2 + 2) \sin x \) is \( y’ = 14x \sin x + (7x^2 + 2) \cos x \). This result encapsulates the application of the product rule and highlights the importance of understanding differentiation techniques in A Level Maths. The expression combines both sine and cosine functions, reflecting the interplay between polynomial and trigonometric functions in calculus.