Find \(y^{\prime}\) where \(y\) is given by (a) \(\frac{x^{2}+1}{2 x+3}\) (b) \(\frac{\cos 2 t}{\sin 3 t}\) (c) \(\frac{r^{3}}{3 e^{2 r}}\) (d) \(\frac{v+\sin v}{1+\mathrm{e}^{v}}\) (e) \(\frac{\sqrt{x}}{x+1}\)

The quotient rule is a technique used in calculus for finding the derivative of a function that is the ratio of two differentiable functions. When we have a function in the form of \( \frac{u}{v} \) where \( u \) and \( v \) are functions of \( x \) or another variable, and \( v \) is not zero, the quotient rule can be applied to find the derivative of this quotient function.

As shown in the exercise, the quotient rule formula is given by: \[ \left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2} \
The process involves three major steps:

• Computing the derivative of the numerator \( u' \) and the denominator \( v' \) separately.
• Plugging \( u' \) and \( v' \) into the formula to calculate the derivative of the quotient.
• Simplifying the result to get the final form of the derivative.

The examples in the exercise show how to apply the quotient rule to different functions.

A derivative in calculus represents the rate at which a function is changing at any given point. It's a fundamental concept used to understand the behavior of functions, including their increasing or decreasing nature, extremum points, and the concavity of their graphs.

When finding derivatives, it's essential to recall the basic derivative rules, such as the power rule \( (x^n)' = nx^{n-1} \) and the derivatives of trigonometric functions, exponential functions, and logarithms.

For example, in the textbook exercise, the derivative of \( x^2 + 1 \) using the power rule is \( 2x \) and the derivative of \( 2x + 3 \) is simply \( 2 \) because the derivative of a constant term is zero. Understanding these fundamental principles is key to applying more complex rules like the quotient rule or the chain rule.

The chain rule is another vital tool in calculus, used to differentiate composite functions. A composite function is a function made by combining two or more functions in a way that the output of one function becomes the input of another.

The rule is defined as: If \( h(x) = f(g(x)) \) then the derivative of \( h \) with respect to \( x \) is \( h'(x) = f'(g(x)) \cdot g'(x) \.

In essence, you find the derivative of the outer function and multiply it by the derivative of the inner function. While the given exercise does not include an explicit use of the chain rule, it's crucial to recognize when this rule is applicable, such as in part (d) when differentiating \( e^v \: the inside function \( v \) also needs to be differentiated according to the chain rule.