Salinas Wheeler
10/20/2023 · Middle School
Use the quotient rule to find the derivative of \[ \frac{-9 \sin (x)-1}{3 x^{8}+6} \]
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Step-by-step Solution
Find the derivative with respect to \( x \) for \( \frac{-9 \sin (x)-1}{3 x^{8}+6} \).
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dx}\left(\frac{-9\sin\left(x\right)-1}{3x^{8}+6}\right)\)
- step1: Rewrite the expression:
\(\frac{d}{dx}\left(-\frac{9\sin\left(x\right)+1}{3x^{8}+6}\right)\)
- step2: Simplify:
\(\frac{d}{dx}\left(\left(-9\sin\left(x\right)-1\right)\times \frac{1}{3}\left(x^{8}+2\right)^{-1}\right)\)
- step3: Rewrite the expression:
\(\frac{d}{dx}\left(-\frac{9\sin\left(x\right)+1}{3\left(x^{8}+2\right)}\right)\)
- step4: Use differentiation rules:
\(-\frac{\frac{d}{dx}\left(9\sin\left(x\right)+1\right)\times 3\left(x^{8}+2\right)-\left(9\sin\left(x\right)+1\right)\times \frac{d}{dx}\left(3\left(x^{8}+2\right)\right)}{\left(3\left(x^{8}+2\right)\right)^{2}}\)
- step5: Calculate:
\(-\frac{9\cos\left(x\right)\times 3\left(x^{8}+2\right)-\left(9\sin\left(x\right)+1\right)\times 24x^{7}}{\left(3\left(x^{8}+2\right)\right)^{2}}\)
- step6: Calculate:
\(-\frac{27\cos\left(x\right)\left(x^{8}+2\right)-24\left(9\sin\left(x\right)+1\right)x^{7}}{9x^{16}+36x^{8}+36}\)
- step7: Calculate:
\(-\frac{9\cos\left(x\right)\left(x^{8}+2\right)-8\left(9\sin\left(x\right)+1\right)x^{7}}{3x^{16}+12x^{8}+12}\)
- step8: Simplify:
\(-\frac{9\cos\left(x\right)\left(x^{8}+2\right)+\left(-72\sin\left(x\right)-8\right)x^{7}}{3x^{16}+12x^{8}+12}\)
- step9: Multiply the terms:
\(-\frac{9x^{8}\cos\left(x\right)+18\cos\left(x\right)+\left(-72\sin\left(x\right)-8\right)x^{7}}{3x^{16}+12x^{8}+12}\)
The derivative of the given function \[ \frac{-9 \sin (x)-1}{3 x^{8}+6} \] using the quotient rule is \[ -\frac{9x^{8}\cos(x)+18\cos(x)+(-72\sin(x)-8)x^{7}}{3x^{16}+12x^{8}+12} \]
Quick Answer
The derivative is \[ -\frac{9x^{8}\cos(x)+18\cos(x)+(-72\sin(x)-8)x^{7}}{3x^{16}+12x^{8}+12} \]
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