A. Find the slope of the tangent line to the graph of f at...

a. Find the slope of the tangent line to the graph of $f$ at the given point.
b. Find the slope-intercept equation of the tangent line to the graph of $f$ at the given point.
$$f(x)=\sqrt{x} \text { at }(16,4)$$

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Key Concepts

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its variable. It provides the slope of the tangent line to the graph at any given point, making it a fundamental concept in calculus for analyzing how functions change.

Differentiation of Radical Functions

Differentiating functions containing radicals, such as square root functions, often involves rewriting the radical as an exponent (for instance, x^(1/2)) and then applying the power rule. This process yields a new function that gives the slope of the tangent at any point on the original curve.

Tangent Line

The tangent line to a curve at a particular point locally approximates the function by having the same instantaneous rate of change at that point. Finding the tangent involves computing the derivative at the point of interest and using this slope along with the point to define the line.

Slope-Intercept Form

The slope-intercept form of a line is expressed as y = mx + b, where m is the slope and b is the y-intercept. This form is useful for quickly identifying both the slope of the line and its intersection with the y-axis after the slope has been determined from the derivative.

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