Question - Solving a Derivative of a Product of Binomial Expressions

To find the derivative of \( y = (7 + x)(7 - x)^2 \), we need to apply the product rule, which states that if \( y = u \cdot v \), then the derivative \( y' = u' \cdot v + u \cdot v' \). Let \( u = 7 + x \) and \( v = (7 - x)^2 \).

First, we find the derivative of \( u \) with respect to \( x \): \( u' = \frac{d}{dx}(7 + x) = 0 + 1 = 1 \).

Next, we find the derivative of \( v \) with respect to \( x \). For this, we can use the chain rule since \( v \) is a function raised to a power: \( v = (7 - x)^2 \), let \( w = 7 - x \), then \( v = w^2 \).

Derivative of \( w \) with respect to \( x \) is: \( w' = \frac{d}{dx}(7 - x) = 0 - 1 = -1 \).

Now, apply the chain rule to find the derivative of \( v \): \( v' = 2(w)^{2-1} \cdot w' = 2(7 - x) \cdot -1 = -2(7 - x) \).

Finally, applying the product rule: \( y' = u' \cdot v + u \cdot v' = 1 \cdot (7 - x)^2 + (7 + x) \cdot -2(7 - x) \). Simplify the expression to get the final result: \( y' = (7 - x)^2 - 2(7 + x)(7 - x) \).

For further simplification, \( y' \) can be expanded by squaring the binomial and applying the distributive property.