Q.Differentiate with respect to $x$: $(2x - 7)^2 (3x + 5)^3$.

The differentiation of the given function requires the application of the Product Rule, as it is a product of two functions, each of which is a composite function requiring the Chain Rule.

Let $y = (2x - 7)^2 (3x + 5)^3$.

Let $u = (2x - 7)^2$ and $v = (3x + 5)^3$.

1. Differentiate $u$ with respect to $x$ using the Chain Rule:

   $\frac{du}{dx} = 2(2x - 7)^{2-1} \cdot \frac{d}{dx}(2x - 7) = 2(2x - 7)(2) = 4(2x - 7)$.

2. Differentiate $v$ with respect to $x$ using the Chain Rule:

   $\frac{dv}{dx} = 3(3x + 5)^{3-1} \cdot \frac{d}{dx}(3x + 5) = 3(3x + 5)^2(3) = 9(3x + 5)^2$.

3. Apply the Product Rule:

   $\frac{dy}{dx} = (2x - 7)^2 [9(3x + 5)^2] + (3x + 5)^3 [4(2x - 7)]$. …

When we need to differentiate a function that is a product of two other functions, say $y = u(x) \cdot v(x)$, the standard approach is to use the Product Rule. In this specific problem, each of the functions $u(x) = (2x - 7)^2$ and $v(x) = (3x + 5)^3$ is itself a composite function (a function raised to a power), meaning we will also need to apply the Chain Rule when finding $u'(x)$ and $v'(x)$.

The Product Rule allows us to break down the differentiation of a complex product into simpler parts. It essentially says that the rate of change of a product is the rate of change of the first function times the second function, plus the first function times the rate of change of the second function.

Here's how we apply this to the given problem:

1. Identify the two functions in the product.

   Let $y = (2x - 7)^2 (3x + 5)^3$.

   We can define $u = (2x - 7)^2$ and $v = (3x + 5)^3$.

   Our goal is to find $\frac{dy}{dx}$.

2. Recall the Product Rule.

   If $y = u \cdot v$, then $\frac{dy}{dx} = u'v + uv'$.

   Here, $u'$ denotes $\frac{du}{dx}$ and $v'$ denotes $\frac{dv}{dx}$.

3. Differentiate $u$ with respect to $x$.

   $u = (2x - 7)^2$. This is a composite function of the form $f(g(x))$, where $f(x) = x^2$ and $g(x) = 2x - 7$.

   We use the Chain Rule: $\frac{du}{dx} = f'(g(x)) \cdot g'(x)$.

   $f'(x) = 2x$, so $f'(g(x)) = 2(2x - 7)$.

   $g'(x) = \frac{d}{dx}(2x - 7) = 2$.

   Therefore, $u' = 2(2x - 7) \cdot 2 = 4(2x - 7)$.

4. Differentiate $v$ with respect to $x$.

   $v = (3x + 5)^3$. This is also a composite function, with $f(x) = x^3$ and $g(x) = 3x + 5$.

   Using the Chain Rule: $\frac{dv}{dx} = f'(g(x)) \cdot g'(x)$.

   $f'(x) = 3x^2$, so $f'(g(x)) = 3(3x + 5)^2$.

   $g'(x) = \frac{d}{dx}(3x + 5) = 3$.

   Therefore, $v' = 3(3x + 5)^2 \cdot 3 = 9(3x + 5)^2$. …