We are given the function:

$f(x) = (7x^2 + 2x + 5)^{3/2}$

To find the derivative, we will apply the chain rule, which states that if $f(x) = g(h(x))$, then:

$f'(x) = g'(h(x)) \cdot h'(x)$

In this case, $g(u) = u^{3/2}$ and $h(x) = 7x^2 + 2x + 5$. We first differentiate each part:

1. Differentiate $g(u) = u^{3/2}$:

$g'(u) = \frac{3}{2} u^{1/2}$

2. Differentiate $h(x) = 7x^2 + 2x + 5$:

$h'(x) = 14x + 2$

Now apply the chain rule:

$f'(x) = \frac{3}{2} (7x^2 + 2x + 5)^{1/2} \cdot (14x + 2)$

Finally, express the square root using fractional exponents:

$f'(x) = \frac{3}{2} (7x^2 + 2x + 5)^{1/2} (14x + 2)$

This is the derivative of the given function using positive and fractional exponents.

Would you like further explanation or additional questions?

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Related questions:

1. What is the chain rule and how does it apply to composite functions?
2. How do we differentiate functions involving square roots?
3. What is the power rule and how can we apply it to fractional exponents?
4. How do we simplify a derivative expression with a fractional exponent?
5. How would the derivative change if the exponent were negative or a different fraction?

Tip: When dealing with fractional exponents, remember that $x^{a/b} = \sqrt[b]{x^a}$. You can always rewrite fractional exponents as radicals to better visualize the problem.