For the following functions, a. use Equation 3.4 to find the...

For the following functions,
a. use Equation 3.4 to find the slope of the tangent line $m_{\text { tan }}=f^{\prime}(a),$ and
b. find the equation of the tangent line to $f$ at $x=a$ .
$$
f(x)=\sqrt{x+8}, a=1
$$

Key Concepts

Tangent Line Equation

The equation of the tangent line to a function at a specific point uses the derivative to determine its slope. The tangent line represents the best linear approximation of the function near that point. Its equation is typically expressed using the point-slope form, where the slope is the derivative evaluated at the point of tangency, and the line passes through the coordinates of that point.

Point-Slope Form

The point-slope form is a method used to express the equation of a straight line. It is given by the formula y - y? = m(x - x?), where (x?, y?) is a point on the line and m is the slope. This form is particularly useful when the slope of the tangent line at a specific point is known, allowing for the direct formulation of the tangent line's equation.

Derivative

The derivative of a function quantifies the rate at which the function’s value changes with respect to changes in its input. It is defined as the limit of the difference quotient as the increment in the input approaches zero. This concept is fundamental in calculus because it gives the slope of the tangent line to the function at any given point, thus providing a linear approximation of the function near that point.

Chain Rule

The chain rule is a method in calculus used to differentiate composite functions. When a function is composed of two or more functions, the chain rule provides a means to compute the derivative by taking the derivative of the outer function evaluated at the inner function and then multiplying it by the derivative of the inner function. This rule is critical when dealing with functions where one function is nested within another.

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