Final answer:
To differentiate the function y = x² + 8x + 3/√x, we use the power rule on each term separately to find that the derivative is 2x + 8 - 3/2·x^(-3/2).
Step-by-step explanation:
To differentiate the function y = x² + 8x + \frac{3}{√x}, we need to apply the rules of differentiation to each term separately. The function is a composition of polynomial and rational functions, thus necessitating the use of the power rule, the constant multiple rule, and the quotient rule where appropriate.
The first term, x², is a basic power of x, and its derivative is obtained using the power rule: if f(x) = x^n, then f'(x) = nx^(n-1). Therefore, the derivative of x² is 2x.
The second term, 8x, is a linear term, and its derivative is simply the coefficient, which is 8.
The third term involves a fractional exponent: \frac{3}{√x} can be rewritten as 3x^(-1/2). Using the power rule again, we find the derivative to be -3/2 · x^(-3/2).
Putting these results together without simplifying, the derivative of the function is 2x + 8 - \frac{3}{2}x^(-3/2), which corresponds to the sum of the derivatives of the individual terms.