Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = √3x; a= 12

Verified step by step guidance

1. Step 1: Identify the function f(x) = \sqrt{3x}. This is a composition of functions, where the outer function is the square root and the inner function is 3x.
2. Step 2: To find the derivative f'(x), use the chain rule. The chain rule states that if you have a composite function f(g(x)), the derivative is f'(g(x)) * g'(x).
3. Step 3: Differentiate the outer function \sqrt{u} with respect to u, which is \frac{1}{2\sqrt{u}}. Here, u = 3x.
4. Step 4: Differentiate the inner function 3x with respect to x, which is 3.
5. Step 5: Apply the chain rule: f'(x) = \frac{1}{2\sqrt{3x}} * 3. Simplify this expression to find f'(x), and then evaluate it at a = 12 to find f'(12).

Key Concepts

Derivatives

A derivative represents the rate of change of a function with respect to its variable. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In practical terms, the derivative at a point gives the slope of the tangent line to the function at that point.

Tangent Lines

A tangent line to a curve at a given point is a straight line that touches the curve at that point without crossing it. The slope of the tangent line is equal to the derivative of the function at that point. This concept is crucial for understanding how functions behave locally around specific values.

Function Evaluation

Function evaluation involves substituting a specific value into a function to determine its output. In the context of derivatives, evaluating the function at a point helps in calculating the derivative at that point. For example, to find f′(a), one must first evaluate the function f(x) at x = a.