Questions: Find y' by (a) applying the Product Rule and (b) multiplying the factors to produce a sum of simpler terms to differentiate. y=(7 x^2+5)(x+4+5/x) a. Apply the Product Rule. Let u=(7 x^2+5) and v=(x+4+5/x). d/dx(u v)=(7 x^2+5)(1-5/x^2)+(x+4+5/x)(14 x) b. Multiply the factors of the original expression, u and v, to produce a sum of simpler terms. y=

Solution Steps

To solve this problem, we need to find the derivative \( y' \) of the given function \( y = (7x^2 + 5)(x + 4 + \frac{5}{x}) \) using two different methods: the Product Rule and by simplifying the expression first.

1. Identify the two functions: \( u = 7x^2 + 5 \) and \( v = x + 4 + \frac{5}{x} \).
2. Use the Product Rule: \( \frac{d}{dx}(uv) = u'v + uv' \).
3. Calculate the derivatives \( u' \) and \( v' \).
4. Substitute these derivatives into the Product Rule formula to find \( y' \).

1. Expand the product \( (7x^2 + 5)(x + 4 + \frac{5}{x}) \) to get a sum of simpler terms.
2. Differentiate the resulting polynomial term by term to find \( y' \).

To find \( y' \) using the Product Rule, we first identify the functions: \[ u = 7x^2 + 5, \quad v = x + 4 + \frac{5}{x} \] Next, we compute the derivatives: \[ u' = 14x, \quad v' = 1 - \frac{5}{x^2} \] Using the Product Rule: \[ y' = u'v + uv' = 14x\left(x + 4 + \frac{5}{x}\right) + \left(1 - \frac{5}{x^2}\right)(7x^2 + 5) \]

We expand the product \( y = (7x^2 + 5)\left(x + 4 + \frac{5}{x}\right) \): \[ y = (7x^2 + 5)\left(x + 4 + \frac{5}{x}\right) = (7x^2 + 5)\left(\frac{x(x + 4) + 5}{x}\right) = \frac{(7x^2 + 5)(x^2 + 4x + 5)}{x} \] Now, we differentiate the simplified expression: \[ y' = 14x(x + 4) + 70 + \frac{(2x + 4)(7x^2 + 5)}{x} - \frac{(7x^2 + 5)(x^2 + 4x + 5)}{x^2} \]

Final Answer