Answer:
See below.
Step-by-step explanation:
First, notice that this is a composition of functions. For instance, let's let [tex]f(x)=\sqrt{x}[/tex] and [tex]g(x)=7x^2+4x+1[/tex]. Then, the given equation is essentially [tex]f(g(x))=\sqrt{7x^2+4x+1}[/tex]. Thus, we can use the chain rule.
Recall the chain rule: [tex]\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)[/tex]. So, let's find the derivative of each function:
[tex]f(x)=\sqrt{x}=x^\frac{1}{2}[/tex]
We can use the Power Rule here:
[tex]f'(x)=(x^\frac{1}{2})'=\frac{1}{2}(x^{-\frac{1}{2} })=\frac{1}{2\sqrt{x}}[/tex]
Now:
[tex]g(x)=7x^2+4x+1[/tex]
Again, use the Power Rule and Sum Rule
[tex]g'(x)=(7x^2+4x+1)'=(7x^2)'+(4x)'+(1)'=14x+4[/tex]
Now, we can put them together:
[tex]y'=f'(g(x))\cdot g'(x)=\frac{1}{2\sqrt{g(x)}} \cdot (14x+4)[/tex]
[tex]y'=\frac{14x+4}{2\sqrt{7x^2+4x+1} } =\frac{7x+2}{\sqrt{7x^2+4x+1} }[/tex]
