use the quotient rule to find the derivative of\n\\( \\frac { - 4 \\sin ( x ) - 6 } { 5 x ^ { 8 } + 2 }…

use the quotient rule to find the derivative of\n\\( \\frac { - 4 \\sin ( x ) - 6 } { 5 x ^ { 8 } + 2 } \\)\nyou do not need to expand out your answer. be careful with parentheses!\nquestion help: video message instructor\nsubmit question jump to answer

use the quotient rule to find the derivative of\n\\( \\frac { - 4 \\sin ( x ) - 6 } { 5 x ^ { 8 } + 2 } \\)\nyou do not need to expand out your answer. be careful with parentheses!\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Identify (u) and (v)

Let (u=-4\sin(x)-6), (v = 5x^{8}+2)

Step2: Find (u') and (v')

Using derivative rules:

  • The derivative of (\sin(x)) is (\cos(x)), and the derivative of a constant is (0). So (u'=-4\cos(x))
  • Using the power rule ((x^{n})'=nx^{n - 1}), for (v = 5x^{8}+2), (v'=5\times8x^{7}=40x^{7})

Step3: Apply the quotient rule

The quotient rule is (\left(\frac{u}{v}\right)'=\frac{u'v - uv'}{v^{2}}) Substitute (u), (u'), (v), (v') into the quotient rule: [ \begin{align*} \frac{(-4\cos(x))(5x^{8}+2)-(-4\sin(x)-6)(40x^{7})}{(5x^{8}+2)^{2}} \end{align*} ]

Answer:

(\frac{(-4\cos(x))(5x^{8}+2)-(-4\sin(x)-6)(40x^{7})}{(5x^{8}+2)^{2}})