Derivatives and tangent lines
a. For the following functions and values of a, find f′(a).
f(x) = 1/ √x; a= 1/4

Verified step by step guidance

1. Step 1: Identify the function and the point of interest. The function given is \( f(x) = \frac{1}{\sqrt{x}} \) and we need to find the derivative at \( a = \frac{1}{4} \).
2. Step 2: Rewrite the function in a form that is easier to differentiate. Express \( f(x) = \frac{1}{\sqrt{x}} \) as \( f(x) = x^{-1/2} \).
3. Step 3: Use the power rule for differentiation. The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \). Apply this to \( f(x) = x^{-1/2} \) to find \( f'(x) \).
4. Step 4: Differentiate \( f(x) = x^{-1/2} \) using the power rule. This gives \( f'(x) = -\frac{1}{2}x^{-3/2} \).
5. Step 5: Evaluate the derivative at \( x = \frac{1}{4} \). Substitute \( x = \frac{1}{4} \) into \( f'(x) = -\frac{1}{2}x^{-3/2} \) to find \( f'(\frac{1}{4}) \).

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 and is often used in optimization problems.

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 the point a before applying derivative rules.