Derivatives and tangent lines a. For the following functions...

Derivatives and tangent lines
a. For the following functions and points, find $f^{\prime}(a)$.
b. Determine an equation of the line tangent to the graph of $f$ at $(a, f(a))$ for the given value of $a$.
$$f(x)=\sqrt{2 x+1} ; a=4$$

Key Concepts

Tangent Line

A tangent line to a curve at a given point is a straight line that just touches the curve at that point, having the same instantaneous slope as the curve. The concept of a tangent line is central in calculus for approximating the behavior of functions near a point and in understanding local linear approximations.

Point-Slope Form

The point-slope form is a way of expressing the equation of a line when a point on the line and the line's slope are known. It is written as y - y? = m(x - x?) and is particularly useful for writing the equation of a tangent line once the derivative (slope) at the point of tangency has been determined.

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its independent variable. It provides a measure of how the function's value changes as its input changes and is interpreted geometrically as the slope of the tangent line to the graph at any given point.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composed function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function, making it invaluable for functions like those involving square roots or other non-linear transformations.

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