Using the product rule when differentiating I am asked to find $dy/dx$ if $y = x(2x + 5)^7$ using the product rule. 

I have made $U = X$ and differentiated this to get $1$.

And $V = (2x + 5)^7$ and differentiated to get $7(2x + 5)^6$.

However the solutions say that $V'$ should be $14(2x + 5)^6$.

Should it be $7$ and not $14$ ? Has there been an error in the solutions or am I incorrect ? 

By the way I got $U$ and $V$ from:
$$
\frac{dy}{dx} = U \cdot \frac{dV}{dx} + V \cdot \frac{dU}{dx}
$$
The problem here, is that you have not used the chain rule.

Put simply, the chain rule says:

$\text{ If you have a function composed of another function }$

$\text{ you must take derivative of outside function $\times$ derivative of inside function }$

$$\frac{d}{dx}\left[ f(g(x))\right] = f'(g(x)) \cdot g'(x)$$



In our case here, we have $(2x+5)^7$. What you must recognize is that this is a combination of $(2x+5)$ and $x^7$.

So we get that the derivative of $(2x+5)^7$ is:

$$7*(2x+5)^6 \text { (product rule on the outside $x^7$ function)} \times (2) \text {(linear differentiation of 2x + 5)} $$

$$14(2x+5)^6$$
You can not apply the reasoning that $\frac{d}{dx}(2x+5)^7 = 7(2x+5)^6$. You have attempted to apply the power rule to a function that is not a polynomial term, and the power rule applies only to functions of the form $ax^n$. Instead, you may do two things:


Expand and apply power rule to each individual term (slow)
Apply the chain rule


The latter of these options you may not have been introduced to you, so here it is:

$$\frac{d}{dx}\left[ f(g(x))\right] = f'(g(x)) \cdot g'(x)$$