Question

attempt 1: 10 attempts remaining. if ( f(x) = \frac{x^3 - 7x}{7x^2 + 9} ), find ( f(x) ). ( f(x) = ) submit answer next item

Explanation:

Step1: Identify the quotient rule

The function \( f(x) = \frac{u}{v} \) where \( u = x^3 - 7x \) and \( v = 7x^2 + 9 \). The quotient rule states that \( f'(x) = \frac{u'v - uv'}{v^2} \).

Step2: Find \( u' \)

Differentiate \( u = x^3 - 7x \) with respect to \( x \). Using the power rule, \( u' = 3x^2 - 7 \).

Step3: Find \( v' \)

Differentiate \( v = 7x^2 + 9 \) with respect to \( x \). Using the power rule, \( v' = 14x \).

Step4: Apply the quotient rule

Substitute \( u \), \( u' \), \( v \), and \( v' \) into the quotient rule formula:
\[
f'(x) = \frac{(3x^2 - 7)(7x^2 + 9) - (x^3 - 7x)(14x)}{(7x^2 + 9)^2}
\]

Step5: Expand the numerator

First, expand \( (3x^2 - 7)(7x^2 + 9) \):
\[
(3x^2)(7x^2) + 3x^2(9) - 7(7x^2) - 7(9) = 21x^4 + 27x^2 - 49x^2 - 63 = 21x^4 - 22x^2 - 63
\]
Next, expand \( (x^3 - 7x)(14x) \):
\[
x^3(14x) - 7x(14x) = 14x^4 - 98x^2
\]
Now, subtract the two expanded expressions:
\[
(21x^4 - 22x^2 - 63) - (14x^4 - 98x^2) = 21x^4 - 22x^2 - 63 - 14x^4 + 98x^2 = 7x^4 + 76x^2 - 63
\]

Step6: Write the final derivative

Substitute the expanded numerator back into the formula:
\[
f'(x) = \frac{7x^4 + 76x^2 - 63}{(7x^2 + 9)^2}
\]

Answer:

\( \boxed{\dfrac{7x^4 + 76x^2 - 63}{(7x^2 + 9)^2}} \)