Final answer:
To find the derivative of f(x) = √(3x + 28) using the definition of the derivative, we need to use the limit definition of the derivative. The derivative is the limit of the difference quotient as h approaches 0, where the difference quotient is (f(x + h) - f(x)) / h. By simplifying the difference quotient and taking the limit, we find that the derivative of f(x) = √(3x + 28) is f'(x) = 3 / (2√(3x + 28)).
Step-by-step explanation:
To find the derivative of f(x) = √(3x + 28) using the definition of the derivative, we need to use the limit definition of the derivative. The derivative is the limit of the difference quotient as h approaches 0, where the difference quotient is (f(x + h) - f(x)) / h.
So, let's start by evaluating the difference quotient:
(f(x + h) - f(x)) / h = (√(3(x + h) + 28) - √(3x + 28)) / h
Next, we need to simplify this expression. We can use the difference of squares formula to simplify the numerator:
(√(3(x + h) + 28) - √(3x + 28)) / h = ((3(x + h) + 28) - (3x + 28)) / (h(√(3(x + h) + 28) + √(3x + 28))))
Finally, we can simplify further and cancel out the common terms:
((3(x + h) + 28) - (3x + 28)) / (h(√(3(x + h) + 28) + √(3x + 28)))) = 3 / (√(3(x + h) + 28) + √(3x + 28)))
Now, take the limit as h approaches 0:
lim(h → 0) 3 / (√(3(x + h) + 28) + √(3x + 28))) = 3 / (2√(3x + 28))
Therefore, the derivative of f(x) = √(3x + 28) is f'(x) = 3 / (2√(3x + 28)).