Questions: Find the derivative of f(w)=w (9/10)+5 w. What is the domain of f'(w) ? Enclose numerators and denominators in parentheses. For example (a-b) /(1+n). Enter the domain in interval notation. For infinity, type infinity. f'(w)= Domain of f'(w)

Solution Steps

The given function is: \[ f(w) = w \cdot \frac{9}{10} + 5w. \] This can be rewritten as: \[ f(w) = \frac{9}{10}w + 5w. \]

Combine the terms involving \( w \): \[ f(w) = \left( \frac{9}{10} + 5 \right) w. \] Simplify the coefficient: \[ \frac{9}{10} + 5 = \frac{9}{10} + \frac{50}{10} = \frac{59}{10}. \] Thus, the function becomes: \[ f(w) = \frac{59}{10}w. \]

The derivative of \( f(w) \) with respect to \( w \) is: \[ f^{\prime}(w) = \frac{d}{dw} \left( \frac{59}{10}w \right). \] Since the derivative of \( w \) is 1, we have: \[ f^{\prime}(w) = \frac{59}{10}. \]

The derivative \( f^{\prime}(w) = \frac{59}{10} \) is a constant function. The domain of a constant function is all real numbers. Therefore, the domain of \( f^{\prime}(w) \) is: \[ (-\infty, \infty). \]

Final Answer