Cruz Ray
07/14/2024 · Middle School
Use the quotient rule to find the derivative of \[ \frac{-9 \sin (x)-1}{3 x^{8}+6} \] You do not need to expand out your answer. Be careful with parentheses!
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Step-by-step Solution
To find the derivative of the function
\[
y = \frac{-9 \sin (x) - 1}{3 x^{8} + 6}
\]
we will use the quotient rule. The quotient rule states that if you have a function in the form \( \frac{u}{v} \), where \( u \) and \( v \) are both functions of \( x \), then the derivative \( y' \) is given by:
\[
y' = \frac{u'v - uv'}{v^2}
\]
In our case, we have:
- \( u = -9 \sin(x) - 1 \)
- \( v = 3x^8 + 6 \)
Now, we need to find the derivatives \( u' \) and \( v' \).
1. **Finding \( u' \)**:
\[
u' = \frac{d}{dx}(-9 \sin(x) - 1) = -9 \cos(x)
\]
2. **Finding \( v' \)**:
\[
v' = \frac{d}{dx}(3x^8 + 6) = 24x^7
\]
Now we can apply the quotient rule:
\[
y' = \frac{u'v - uv'}{v^2}
\]
Substituting \( u \), \( u' \), \( v \), and \( v' \) into the formula:
\[
y' = \frac{(-9 \cos(x))(3x^8 + 6) - (-9 \sin(x) - 1)(24x^7)}{(3x^8 + 6)^2}
\]
Thus, the derivative of the given function is:
\[
y' = \frac{-9 \cos(x)(3x^8 + 6) + (9 \sin(x) + 1)(24x^7)}{(3x^8 + 6)^2}
\]
This is the final answer, and it is expressed in a compact form without expanding the terms.
Quick Answer
The derivative of the function is:
\[
y' = \frac{-9 \cos(x)(3x^8 + 6) + (9 \sin(x) + 1)(24x^7)}{(3x^8 + 6)^2}
\]
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