Use differentials to determine the approximate change in the...

Use differentials to determine the approximate change in the value of the given function as its argument changes from the given value by the given amount. What is the approximate value of the function after the change?
$f(x)=\sqrt{3 x+1},$ as $x$ increases from 1 to 1.08

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Key Concepts

Derivative

The derivative of a function represents the rate at which the function's value changes with respect to changes in its input. It provides the slope of the tangent line at a particular point and is central to forming linear approximations of the function.

Chain Rule

The chain rule is a method for finding the derivative of composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function multiplied by the derivative of the inner function. This is particularly important when differentiating functions that are composed together, such as a square root function involving a linear expression.

Differentials

Differentials refer to the small changes in variables and their effects on functions. In calculus, the differential of a function is the product of its derivative and the small change in the input. This concept is used to approximate changes in functions when the input changes by a small amount.

Linear Approximation

Linear approximation uses the idea of differentials to estimate the value of a function near a specific point by using the tangent line at that point. This approximation is useful when the change in the input is small, allowing the function's behavior to be approximated by a straight line through the point.

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