91Ó°ÊÓ

The quotient rule is a method for finding the derivative of a function that is the division of two differentiable functions. When dealing with the function of the form \( y = \frac{u(x)}{v(x)} \), the derivative \( y' \) is given by the formula: \[ y' = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \] where \( u(x) \) and \( v(x) \) are both functions of \( x \), and \( u'(x) \) and \( v'(x) \) are their respective derivatives.

• The numerator of this formula is the difference between two terms: the derivative of the numerator function, \( u(x) \), multiplied by the denominator, \( v(x) \), minus the numerator times the derivative of the denominator.
• Meanwhile, the denominator is simply the square of the denominator function \( v(x) \).

The quotient rule is especially useful when the numerator and denominator are not easily separable for simple integration or differentiation.

Derivatives are a fundamental concept in differential calculus that measure how a function changes as its input changes. The derivative of a function \( f(x) \), denoted \( f'(x) \) or \( \frac{df}{dx} \), represents the slope of the tangent line to the curve at a given point \( x \).

• The process of finding a derivative is known as differentiation.
• Derivatives allow us to determine the rate at which one quantity changes with respect to another.

Finding the derivative of a function involves applying rules of differentiation such as the power rule, product rule, and quotient rule. In the given exercise, applying the quotient rule to find the derivative of \( y = \frac{3x+5}{x^2+1} \) involves calculating each component’s derivative and applying it within the rule. This derivative is then used to approximate changes in the function with \( dx \).

Differentials provide an approximation of how much a function changes based on a small change in the input. When we have a function \( y = f(x) \) and its derivative \( y' \), the differential \( dy \) is defined as \[ dy = y'(x) \, dx \] where \( dx \) is a small change in \( x \).

• To compute \( dy \), you first find the derivative \( y'(x) \) of the function.
• Then multiply it by the small change \( dx \).

In our particular exercise, we calculate \( dy \) using \( dx = 0.2 \) at the point \( x = 1 \) to estimate how much the function changes when \( x \) changes by this small increment. This method is useful because it allows us to estimate changes without recalculating the entire function.

Approximation in calculus often refers to estimating values of a function using its derivative. Differentials provide a linear approximation to a function by using the derivative:

• This approach assumes that changes in \( y \) over small \( x \) increments can be approximated linearly.
• Such techniques are very useful in applied mathematics and sciences where exact values might be difficult or impossible to obtain quickly.

In the original exercise, the differential \( dy \) gives us an approximation that helps predict \( \Delta y \), the actual change in \( y \), more rapidly and with less computation than recalculating the full function. However, \( \Delta y \) (the exact change obtained by computing the function's output at both \( x \) and \( x + \Delta x \)) can often differ from \( dy \) due to non-linear behavior of the function. Thus, differentials are often most accurate when changes in \( x \) are extremely small.