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 $a$.
$$f(x)=\frac{1}{x+5} ; a=5$$

Key Concepts

Derivative

The derivative represents the instantaneous rate of change of a function at a given point. It is defined as the limit of the difference quotient as the increment approaches zero, providing critical information about the function’s behavior and how it changes locally.

Differentiation Rules

Differentiation rules, including the power rule, product rule, quotient rule, and chain rule, are formulas that simplify the process of finding a derivative. These rules help in efficiently computing the derivative of a wide variety of functions without resorting back to the limit definition every time.

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. It allows the derivative of a composite function to be calculated by differentiating the outer function with respect to the inner function and then multiplying by the derivative of the inner function.

Tangent Line

The tangent line to a curve at a specific point is a straight line that touches the curve only at that point and has the same slope as the curve at that point. It is a visual representation of the function’s instantaneous rate of change and is often determined using the derivative at the point of tangency.

Point-Slope Form

The point-slope form is a method for writing the equation of a line when a point on the line and the slope are known. It is written as y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope, making it particularly useful for defining the equation of a tangent line.

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