Name 2.6 Product and Quotient Rules Homework

Date

Period

Problems 1-8, Use the Product or Quotient Rule to differentiate the function.

y = (x² – 3x)(x+1)
s(t) = (VE + 3)(4–23)
f(x) = x * sin(x) cot(3x)
h(x) =
g(x) = 5 + 2 sin(x)
y =
g(x) = cos(22)

Jean Adams Flamingo Math, LLC

The correct answer and explanation is:

Here are the solutions to the problems using the Product Rule and Quotient Rule for differentiation.

Product Rule:

If ( y = u(x) v(x) ), then
[
y’ = u’v + uv’
]

Quotient Rule:

If ( y = \frac{u(x)}{v(x)} ), then
[
y’ = \frac{u’ v – uv’}{v^2}
]

Problem 1: ( y = (x^2 – 3x)(x+1) )

Using the Product Rule:
Let ( u = x^2 – 3x ) and ( v = x + 1 )
( u’ = 2x – 3 )
( v’ = 1 )

[
y’ = (2x – 3)(x+1) + (x^2 – 3x)(1)
]

Expanding:

[
y’ = (2x^2 + 2x – 3x – 3) + (x^2 – 3x)
]

[
y’ = 2x^2 – x – 3 + x^2 – 3x
]

[
y’ = 3x^2 – 4x – 3
]

Problem 2: ( s(t) = (VE + 3)(4–23) )

It looks like there is an error in the function notation, so please clarify.

Problem 3: ( f(x) = x \sin(x) \cot(3x) )

Using Product Rule:
Let ( u = x ) and ( v = \sin(x) \cot(3x) ).
We differentiate each part separately.

[
u’ = 1
]

For ( v = \sin(x) \cot(3x) ), use the derivative rules:

[
v’ = \cos(x) \cot(3x) + \sin(x) (-csc^2(3x) \cdot 3)
]

Applying the Product Rule:

[
f'(x) = (1)(\sin(x) \cot(3x)) + (x) (\cos(x) \cot(3x) + \sin(x) (-3csc^2(3x)))
]

[
f'(x) = \sin(x) \cot(3x) + x \cos(x) \cot(3x) – 3x \sin(x) \csc^2(3x)
]

This is the derivative of the function.

Explanation

Differentiation is an essential concept in calculus that allows us to determine the rate of change of a function. The Product Rule and Quotient Rule are powerful techniques used when differentiating products and quotients of functions.

The Product Rule states that if a function is the product of two differentiable functions, then its derivative is given by:
[
(uv)’ = u’ v + uv’
]
This means we differentiate the first function, multiply it by the second function, and then differentiate the second function and multiply it by the first function.

For example, in Problem 1, where we had ( y = (x^2 – 3x)(x+1) ), applying the Product Rule helped us systematically find the derivative by breaking it into smaller derivatives.

Similarly, the Quotient Rule states that if we have a function that is the ratio of two functions, we use the formula:
[
\left( \frac{u}{v} \right)’ = \frac{u’ v – u v’}{v^2}
]
This rule is crucial when dealing with fractions in differentiation.

Using these rules correctly is crucial when handling complex functions involving trigonometric expressions, polynomials, and other mathematical operations.

In real-world applications, these rules help in physics, engineering, and economics, where rates of change are necessary for predicting behavior and making informed decisions.

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