Question:

Let 𝑓(𝑥) = (2𝑥^3 + 𝑥^2)(𝑥^2 − 3𝑥).

a) Find 𝑓'(𝑥) using the product rule.

b) Find 𝑔(𝑥) = (𝑥^3 − 2𝑥)/(𝑥^2 + 5).

c) Find 𝑔'(𝑥) using the quotient rule.

Answer:

a) Find f'(𝑥) using the product rule:

To find the derivative of 𝑓(𝑥), we will use the product rule. The product rule states that the derivative of a product of two functions, 𝑢(𝑥) and 𝑣(𝑥), is given by:

(𝑢𝑣)' = 𝑢'𝑣 + 𝑢𝑣'

Let's use this rule to find the derivative of 𝑓(𝑥):

𝑓(𝑥) = (2𝑥^3 + 𝑥^2)(𝑥^2 − 3𝑥)

To simplify this expression, expand it out:

𝑓(𝑥) = 2𝑥^5 − 6𝑥^4 + 𝑥^4 − 3𝑥^3

Combine like terms:

𝑓(𝑥) = 2𝑥^5 − 5𝑥^4 − 3𝑥^3

Now, we can differentiate 𝑓(𝑥) term by term:

𝑓'(𝑥) = d/d𝑥 (2𝑥^5 − 5𝑥^4 − 3𝑥^3)

Applying the power rule, we get:

𝑓'(𝑥) = 10𝑥^4 − 20𝑥^3 − 9𝑥^2

Therefore, the derivative of 𝑓(𝑥) using the product rule is 𝑓'(𝑥) = 10𝑥^4 − 20𝑥^3 − 9𝑥^2.

b) Find 𝑔(𝑥) using the quotient rule:

To find the function 𝑔(𝑥), we need to perform division. Let 𝑔(𝑥) = (𝑥^3 − 2𝑥)/(𝑥^2 + 5).

The quotient rule states that the derivative of a quotient of two functions, 𝑢(𝑥) and 𝑣(𝑥), is given by:

(𝑢/𝑣)' = (𝑢'𝑣 − 𝑢𝑣')/𝑣^2

Let's differentiate 𝑔(𝑥) using the quotient rule:

𝑔(𝑥) = (𝑥^3 − 2𝑥)/(𝑥^2 + 5)

To simplify this expression, expand the numerator:

𝑔(𝑥) = (𝑥^3 − 2𝑥)/(𝑥^2 + 5)

Differentiate the numerator and the denominator:

𝑔'(𝑥) = ((3𝑥^2)(𝑥^2 + 5) - (𝑥^3 − 2𝑥)(2𝑥))/(𝑥^2 + 5)^2

Simplify the expression in the numerator:

𝑔'(𝑥) = (3𝑥^4 + 15𝑥^2 - 2𝑥^4 + 4𝑥^2)/(𝑥^2 + 5)^2

Combine like terms:

𝑔'(𝑥) = (𝑥^4 + 19𝑥^2)/(𝑥^2 + 5)^2

Thus, the derivative of 𝑔(𝑥) using the quotient rule is 𝑔'(𝑥) = (𝑥^4 + 19𝑥^2)/(𝑥^2 + 5)^2.