{Use of Tech} Approximating derivatives Assuming the limit exists, the definition of the derivative f′(a) = lim h→0 f(a + h) − f(a) / h implies that if ℎ is small, then an approximation to f′(a) is given by
f' (a) ≈ f(a+h) - f(a) / h. If ℎ > 0 , then this approximation is called a forward difference quotient; if ℎ < 0 , it is a backward difference quotient. As shown in the following exercises, these formulas are used to approximate f′ at a point when f is a complicated function or when f is represented by a set of data points. <IMAGE>
Let f (x) = √x.
a. Find the exact value of f' (4).

Verified step by step guidance

1. Step 1: Recall the definition of the derivative for a function f(x). The derivative f'(x) is defined as the limit: f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
2. Step 2: Identify the function given in the problem. Here, f(x) = \sqrt{x}. We need to find the derivative at x = 4, so we are looking for f'(4).
3. Step 3: Substitute f(x) = \sqrt{x} into the derivative definition. This gives us: f'(4) = \lim_{h \to 0} \frac{\sqrt{4+h} - \sqrt{4}}{h}.
4. Step 4: Simplify the expression under the limit. To do this, multiply the numerator and the denominator by the conjugate of the numerator: \frac{\sqrt{4+h} - 2}{h} \cdot \frac{\sqrt{4+h} + 2}{\sqrt{4+h} + 2}.
5. Step 5: Simplify the resulting expression and evaluate the limit as h approaches 0. This will give you the exact value of f'(4).