Questions: Find the derivative of the function. r(t)=(3 t-5)^7/(9 t^2+5) r'(t)=

Solution Steps

To find the derivative of the function \( r(t) = \frac{(3t-5)^7}{9t^2+5} \), we will use the quotient rule. The quotient rule states that if you have a function \( \frac{u(t)}{v(t)} \), its derivative is given by \( \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \). Here, \( u(t) = (3t-5)^7 \) and \( v(t) = 9t^2+5 \). We will first find the derivatives \( u'(t) \) and \( v'(t) \) using the chain rule and power rule, respectively, and then apply the quotient rule.

We start with the function \( r(t) = \frac{(3t - 5)^7}{9t^2 + 5} \). Here, we define:

• \( u(t) = (3t - 5)^7 \)
• \( v(t) = 9t^2 + 5 \)

Next, we compute the derivatives of \( u(t) \) and \( v(t) \):

• The derivative of \( u(t) \) is given by: \[ u'(t) = 21(3t - 5)^6 \]
• The derivative of \( v(t) \) is: \[ v'(t) = 18t \]

Using the quotient rule, the derivative \( r'(t) \) is calculated as follows: \[ r'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{(v(t))^2} \] Substituting the derivatives and functions: \[ r'(t) = \frac{21(3t - 5)^6(9t^2 + 5) - (3t - 5)^7(18t)}{(9t^2 + 5)^2} \]

The expression for \( r'(t) \) simplifies to: \[ r'(t) = \frac{-18t(3t - 5)^7 + 21(3t - 5)^6(9t^2 + 5)}{(9t^2 + 5)^2} \]

Final Answer