Questions: The derivative of (x^2 + 5)^7 with respect to x is 14x(x^2 + 5)^6.

Solution Steps

To solve this problem, we need to verify if the given derivative is correct. We will use the chain rule to differentiate the function \((x^2 + 5)^7\) and then compare it with the given expression \(14x(x^2 + 5)^6\).

We start with the function \( f(x) = (x^2 + 5)^7 \).

To differentiate \( f(x) \), we use the chain rule. The chain rule states that if \( f(x) = g(h(x)) \), then \( f'(x) = g'(h(x)) \cdot h'(x) \).

Here, let \( u = x^2 + 5 \). Then \( f(x) = u^7 \).

First, differentiate the outer function \( u^7 \) with respect to \( u \): \[ \frac{d}{du}(u^7) = 7u^6 \]

Next, differentiate the inner function \( u = x^2 + 5 \) with respect to \( x \): \[ \frac{d}{dx}(x^2 + 5) = 2x \]

Using the chain rule, we combine the results: \[ \frac{d}{dx}((x^2 + 5)^7) = 7(x^2 + 5)^6 \cdot 2x \]

Simplify the expression: \[ \frac{d}{dx}((x^2 + 5)^7) = 14x(x^2 + 5)^6 \]

Final Answer