Questions: Use the Product Rule or Quotient Rule to find the derivative. g(u) = (7 + 6/u^2)(2u^4 + 3u^2 - 7)

Solution Steps

To find the derivative of the function \( g(u) = \left(7+\frac{6}{u^{2}}\right)\left(2 u^{4}+3 u^{2}-7\right) \), we will use the Product Rule. The Product Rule states that if you have a function \( h(x) = f(x) \cdot k(x) \), then the derivative \( h'(x) \) is given by \( h'(x) = f'(x) \cdot k(x) + f(x) \cdot k'(x) \). Here, we will identify \( f(u) = 7+\frac{6}{u^{2}} \) and \( k(u) = 2 u^{4}+3 u^{2}-7 \), find their derivatives, and then apply the Product Rule.

We start with the function \( g(u) \) defined as: \[ g(u) = \left(7 + \frac{6}{u^2}\right)\left(2u^4 + 3u^2 - 7\right) \] We identify the two parts of the product: \[ f(u) = 7 + \frac{6}{u^2} \quad \text{and} \quad k(u) = 2u^4 + 3u^2 - 7 \]

Next, we compute the derivatives of \( f(u) \) and \( k(u) \): \[ f'(u) = -\frac{12}{u^3} \] \[ k'(u) = 8u^3 + 6u \]

Using the Product Rule, we find the derivative \( g'(u) \): \[ g'(u) = f'(u) \cdot k(u) + f(u) \cdot k'(u) \] Substituting the expressions we have: \[ g'(u) = \left(-\frac{12}{u^3}\right)\left(2u^4 + 3u^2 - 7\right) + \left(7 + \frac{6}{u^2}\right)\left(8u^3 + 6u\right) \]

Now we simplify the expression for \( g'(u) \): \[ g'(u) = \left(7 + \frac{6}{u^2}\right)(8u^3 + 6u) - \frac{12}{u^3}(2u^4 + 3u^2 - 7) \]

Final Answer